SYNCHRONOUS  MOTORS 


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SYNCHRONOUS  MOTORS 
AND  CONVERTERS 


THEORY    AND    METHODS 
OF    CALCULATION    AND   TESTING 

BY 

ANDRE    E.    BLONDEL 

n 

Graduate  and  Professor,  National  School  of  Bridges  and  Highways   of  France;    Member 

of  the  Legion  of  Honor  of  France;   Honorary  Member  of  the  American  Institute 

oj  Electrical  Engineers;    Member  of  Numerous  Scientific,    Technical,  and 

Engineering  Societies  in  France  and  other  European   Countries 


TRANSLATED   FROM  THE   FRENCH 

BY 

C.    O.   MAILLOUX,    M.E.,    M.S. 

Consulting  Electrical  Engineer 

WITH  ADDITIONAL   CHAPTERS 

BY 

COMFORT    A.    ADAMS,   S.B.,    E.E. 

Professor  of  Electrical  Engineering  in    Harvard  University 


McGRAW-HILL   BOOK   COMPANY 

239  WEST  39TH  STREET,  NEW  YORK 

6  BOUVERIE  STREET,  LONDON,  E.  C. 

1913 


1  H 


Engineering 
Library 


COPYRIGHT,  1913, 

BY  THE 

McGRAW-HILL  BOOK  COMPANY 


TRANSLATOR'S   PREFACE 


PROF.  A.  BLONDEL  was  the  first  writer  to  publish  a  systematic, 
comprehensive  work  on  Synchronous  Motors;  and  his  book,  although 
it  has  now  been  before  the  public  several  years,  still  remains  the 
leading  work  on  that  important  subject.  Owing  to  the  non-existence 
of  an  English  edition,  it  is  not,  however,  as  well  known  and  as  much 
appreciated  as  it  deserves  to  be,  by  English-speaking  readers. 

The  translation  of  this  celebrated,  one  might  almost  say  classical, 
work  into  English,  was  undertaken  at  the  suggestion  of  teachers  and 
others  who  were  desirious  of  making  more  extensive  use  of  the  work 
than  is  possible  if  the  French  text  alone  is  available. 

The  author  and  the  publisher  both  accepted  the  suggestion  that 
the  scope  and  the  usefulness  of  the  book  might  be  increased  mate- 
rially by  including  in  it  some  reference  to  "  Rotary  Converters." 
Excellent  material  for  this  purpose  was  already  available  in  the  form 
of  a  paper  presented  by  Prof.  Blondel,  at  the  Electrical  Congress  in 
Paris,  in  1900.  Two  other  papers  presented  by  him  at  the  Electrical 
Congress  at  St.  Louis,  in  1904,  also  were  of  sufficient  interest  in  this 
connection  to  make  their  reproduction  desirable. 

It  was  decided  to  separate  the  contents  of  the  book  into  three 
distinct  parts.  Part  1,  relating  to  Synchronous  Motors,  corresponds 
to  the  original  French  work  on  Synchronous  Motors.  The  author 
himself  corrected  the  French  text,  and  he  also  supplemented  it 
with  much  new  matter  while  the  translation  was  in  process.  The 
proofs  of  the  English  text  were  submitted  to  several  persons  who 
were  well  qualified  to  criticise  the  text  and  suggest  improvements 
therein.  The  French  text  of  Part  I  is  also  supplemented  by  an 
additional  chapter  contributed  by  Prof.  C.  A.  Adams,  of  Harvard 
University.  Part  I  may  therefore  be  considered  fairly  well  brought 
up  to  date.  Part  II,  relating  to  Synchronous  or  Rotary  Converters, 
is  made  up  of  old  and  new  matter.  The  old  matter  (Chapters  I 
and  II),  constitutes  a  translation  of  Prof.  Blondel's  Paris-Congress 


260087 


vi  TRANSLATOR'S   PREFACE 

paper  of  1900.  The  new  matter  consists  of  three  chapters  by  Prof. 
Blondel,  and  a  chapter  contributed  by  Prof.  C.  A.  Adams.  Part  III 
contains  the  reprints  of  the  two  papers  presented  by  Prof.  Blondel 
at  the  St.  Louis  Electrical  Congress  in  1904,  relating  to  the  applica- 
tion of  his  "  two-reaction  "  method  to  alternators. 

A  very  few  modifications  and  additions  have  been  made  in  the 
text  by  the  translator.  His  own  strong  objections  to  the  terms 
"  wattless  "  and  "  watted  "  made  him  very  willing  to  eliminate  them; 
and  the  recent  action  of  the  Standards  Committee  of  the  American 
Institute  of  Electrical  Engineers  in  sanctioning  and  recommending 
"  reactive  "  and  "  active  "  as  substitutes,  furnished  the  incentive 
and  the  pretext  for  doing  this,  even  after  a  considerable  portion  of 
the  type  had  been  set.  This  book  will,  therefore,  be  the  first  book 
in  which  "  wattless  "  and  "  watted  "  are  replaced  by  "  reactive  "  and 
"  active  "  respectively. 

The  translator  desires  to  express  his  gratitude  and  sincere  thanks 
to  all  who  have  encouraged  and  assisted  him  in  the  preparation 
of  this  book  for  the  press.  Special  acknowledgment  should  be  made 
of  the  very  valuable  services  rendered  by  Prof.  C.  A.  Adams.  He 
was  the  first  to  suggest  the  publication  of  this  book,  and  he  has 
made  many  excellent  suggestions,  besides  contributing  two  new 
chapters,  and  also  reading  and  correcting  the  proofs  of  the  entire 
book.  Special  thanks  are  also  due  to  Prof.  E.  J.  Berg,  of  the  Univer- 
sity of  Illinois,  for  his  very  thorough  reading  of  the  proofs  of  Part  I, 
and  for  very  excellent  suggestions  for  changes  and  additions  made 
by  him,  some  of  which  are  incorported  in  notes  inserted  in  the  text, 
and  identified  by  the  initials  "  EJ.B."  Thanks  for  reading  the 
proofs  of  Part  I,  and  making  corrections  or  suggestions  are  due  to 
the  following: 

Prof.  D.  C.  Jackson,  of  Massachusetts  Institute  of  Technology; 
Profs.  W.  I.  Slichter  and  Morton  Arendt,  of  Columbia  University; 
Prof.  H.  H.  Norris,  of  Cornell  University;  Mr.  A.  L.  Jones,  of  the 
General  Electrical  Company. 

Thanks  are  due  to  Mr.  C.  W.  Stone,  of  the  General  Electric  Com- 
pany, and  Mr.  W.  S.  Rugg,  of  the  Westinghouse  Electric  and  Manu- 
facturing Co.,  for  the  data  relating  to  American  Synchronous  Motors, 
given  in  Appendix  B;  and  also  to  the  Edison  Illuminating  Company 
of  Detroit,  Mich.,  and  to  Mr.  A.  A.  Meyer  of  its  engineering  depart- 
ment, for  the  information  relative  to  the  practical  use  of  synchronous 
condensers  contained  in  Appendix  C. 


TRANSLATOR'S  PREFACE  vii 

The  Translator  desires  to  make  special  acknowledgment  of 
the  great  courtesy  and  kindness  of  the  distinguished  author,  Prof. 
Andre  Blondel  himself.  The  preparation  of  this  book  for  the  press 
has  been  a  labor  of  love  which  has  occupied  very  pleasantly  and 
profitably  a  portion  of  the  leisure  moments  of  the  translator.  He 
considers  himself  remunerated  amply,  for  the  work  involved,  by  the 
great  privilege  which  has  been  one  of  the  perquisites  incidental  to 
the  task,  namely,  that  of  closer  personal  contact  and  acquaintance 
with  the  author;  and  he  is  very  glad  to  have  had  the  opportunity, 
through  this  translation,  to  help  make  the  work  and  the  talents  of 
the  author  better  known,  as  they  deserve  to  be;  for,  unquestionably, 
Prof.  Blondel  is  one  of  the  great  productive  workers  of  our  time 
in  pure  and  applied  electrical  science.  His  work,  great  as  it  is  in 
itself,  becomes  really  wonderful  and  phenomenal,  when  the  circum- 
stances under  which  it  has  been  done  are  realized  and  appeciated. 
Though  handicapped  most  unfortunately,  by  protracted  serious 
ill  health  and  physical  suffering,  he  has,  nevertheless,  kept  well  in 
the  front  rank  with  his  more  fortunate  contemporaries  and  colleagues 
in  the  entire  world;  and  he  has  achieved  fame  and  renown  by  great 
mental  powers,  by  wonderful  originality  and  versatility,  not  only 
as  a  scientist,  a  teacher,  and  an  author,  but  also  as  an  inventor,  an 
engineer,  and  an  expert.  The  great  respect  which  is  inspired  by  the 
prodigious  quantity  and  the  superior  quality  of  Prof.  Blondel's  work 
turn  to  absolute  wonder  and  to  profound  admiration,  before  the 
wonderful  activity  and  the  untiring  energy  of  his  highly  gifted, 
well-trained  mind.  No  tribute  of  praise  is  too  great  for  the  work  of 
this  man,  who  is  at  the  same  time  a  genius  and  a  hero,  with  an 
innate  love  of  science  and  a  devotion  to  scientific  progress  which 
uphold  and  uplift  him,  and  urge  him  onward,  quand  mdme,  in  spite 
of  ill-health  and  physical  suffering,  to  new  researches  and  new 
achievements. 

THE  TRANSLATOR. 

NEW  YORK,  December,  1912. 


CONTENTS 


TRANSLATOR'S  PREFACE v 

INTRODUCTION xv 


PART  I 

CHAPTER  I. — GENERAL  PRINCIPLES  OF  SYNCHRONOUS  MOTORS.  . .  -     i 

Construction  —  Experimental  Properties  —  Case  of  Equal  Electro- 
motive Forces — Case  of  Unequal  Electromotive  Forces — Elementary 
Explanation  of  Polyphase  Synchronous  Motors — Elementary  Explana- 
tion of  Single-Phase  Synchronous  Motors — Equations  of  Synchronous 
Motors,  Analytical  Theory — Case  of  Symmetrical  Polyphase  Motors — 
Graphical  Representation  of  Operative  Conditions,  Blakesley's  Method — 
Equation  of  the  Synchronous  Motor  by  the  Method  of  Complex 
Variables — Excitation  of  Synchronous  Motors. 

CHAPTER  II. — DETAILED  STUDY  OF  OPERATION  WITH  NORMAL 

LOAD 31 

I.   PRINCIPLES    OF   THE   ELEMENTARY   THEORY  —    NOTATION   — 
PRINCIPLE  OF  BIPOLAR  DIAGRAMS 

Bipolar  Diagram  of  the  First  Kind — Motor- Vector  E2  taken  as  Fixed 
Axis — Applications  of  the  Diagram  of  the  First  Kind — Line  of  Equal 
Power  Occurring  with  Constant  Excitation — Lines  of  Equal  Phase — 
Limit-Circle  of  Current — Algebraical  Relations  Deduced  from  the 
Diagram — Numerical  Example — Diagram  of  the  Second  Kind.  The  Vec- 
tor of  the  Generator  E.M.F.,  E\,  as  a  Fixed  Axis — Power-Values  as 
Function  of  the  Lag-Angle  0 — Use  of  this  Diagram  'for  the  Study  of 
Different  Loads — Curves  of  Constant  Electric  Power  of  the  Motor 
when  the  Generator  has  Constant  Excitation — Current-Limit  Circle — 
Lines  of  Equal  Phase — Numerical  Example. 

ix 


CONTENTS 

PAGE 

II.    OPERATION  OF  A  MOTOR  WITH  CONSTANT  EXCITATION,  SUP- 
PLIED AT  CONSTAT  E.M.F 49 

Maximum  Power — Means  of  Determining  the  Practical  Stability  of 
Synchronous  Motors— Variations  of  Stability  with  Operating  Conditions- 
Numerical  Example. 


III.  COMPARISON  OF  POSSIBLE  OUTPUTS  AT  CONSTANT  LOAD  WITH 

VARIOUS  EXCITATIONS.   CONSTANT  POTENTIAL  SUPPLY 55 

Existence  of  a  Current-Minimum — V-Curves — Use  of  Diagram  of 
the  First  Kind — Predetermination  of  V-Curves — Theoretical  Form  of 
V-Curves — Curve  of  Reactive  Current — Expression  for  Reactive  Current 
— Comparison  of  Outputs  which  the  Same  Alternator  Can  Develop  with 
the  Same  Terminal  Voltage,  when  used  either  as  a  Generator  or  as  a 
Motor. 


IV.    INFLUENCE  OF  MOTORS  ON  THE  GENERAL  OPERATION  OF  AN 

A.C.  ELECTRICAL  DISTRIBUTION  SYSTEM 64 

Effect  of  Current  of  Synchronous  Motors  on  Distribution  Systems — 
Compensation  with  Respect  to  the  Line  or  Circuit — First  Numerical 
Example — Second  Numerical  Example — Economic  Study  of  Compensa- 
tion for  the  Line  of  Means  of  Motors  Running  without  Lgad — Saving  in 
Cost  of  Equipment — Saving  in  Annual  Operating  Cost — Numerical 
Example — Table  I,  Saving  in  Cost  of  Equipment — Table  II,  Saving  in 
Operating  Cost — Table  III,  Numerical  Example — Economy  of  Compensa- 
tion for  the  Distributing  System  by  Means  of  Synchronous  Motors 
Running  with  Load — Numerical  Example — Regulation  of  Distribution- 
Voltage — Compensation  with  Respect  to  the  Generators — Numerical 
Example — Comparison  between  Synchronous  and  Induction  Motors — 
Use  of  Synchronous  Motors  to  Raise  Power-factor  in  America. 

CHAPTER  III. — ADDITIONS  TO  THE  THEORY.     SECOND  APPROX- 
IMATION        91 

Imperfections  of  the  Theory — Variations  of  Reactance  with  Lag  of 
Current  and  Saturation  of  Fields.  Armature  Reaction — First  Applica- 
tion of  Corrected  Diagram — Determination  of  Reactive  Current  as  a 
Function  of  the  Excitation,  with  Constant  Active  Current — Particular 
Case  where  the  Permeability  of  the  Field-Circuit  is  Constant  and  the 
Two  Reaction-Coefficients  are  Equal — Second  Application.  Operation 
with  Constant  Excitation,  on  Constant-Potential  Supply  System — 
V-Curves — Influence  of  Field-Saturation  on  Stability — Influence  of 
the  Wave-Form  of  E.M.F. — Simplified  Diagrams. 


CONTENTS  xi 

PAGE 

CHAPTER  IV. — OPERATION  or  SYNCHRONOUS  MOTORS.    HUNT- 
ING     106 

Starting  by  Direct  Current — Starting  with  Alternating  Current  by 
Polyphase  Motors — Synchronism — Observations  on  the  E.M.F.  Induced 
in  the  Poles — Accessory  Starting  Apparatus.  Installation  of  Syn- 
chronous Motors — Starting  of  Single-Phase  Machines — Theory  of  Initial 
Synchronizing — Separate  Excitation — Field  Due  to  a  Commutated  Cur- 
rent— Oscillations  of  Synchronous  Motors — Short-Period  Oscillations — 
Damping  of  Oscillations — Long-Period  Oscillations. 


CHAPTER  V. — TESTS  OF  SYNCHRONOUS  MOTORS 


132 


Characteristic     Curves — Measurement    of    Efficiency — Experimental 
Tests — Advantages  and  Disadvantages  of  Synchronous  Motors. 


CHAPTER  VI. — OTHER    MOTORS    OPERATING    SYNCHRONOUSLY 

WITHOUT  DIRECT-CURRENT  EXCITATION 141 

Reaction  Synchronous  Motors — Synchronous  Motor  with  Alternating 
Fields. 


CHAPTER  VII. — BIPOLAR  DIAGRAM  or  THE  SECOND  KIND  IN 

AMPERE-TURNS 147 

Introduction — Diagram  Transformations  E.M.F.  Diagram — Approxi- 
mate Diagram — Extreme  Cases — Mechanical  Analogue — Length  of 
Air-Gap. 


CHAPTER  VIII. — GENERALIZATION  OF  DIAGRAM  FOR  COUPLED 

SYNCHRONOUS  MACHINES  . .  .166 


PART  II 

CHAPTER  I. — GENERAL  DIAGRAMS  DEDUCED  FROM  THE  DIAGRAM 

FOR  SYNCHRONOUS  MOTORS 172 

Introduction — Notation — Generalities.  Reduction  of  all  Armature 
Reactions  to  the  Single  Direct  Reaction — Factors  Determining  the 
Practical  Conditions  of  Operators. 


xii  CONTENTS 

PAGE 

I.   CONDITIONS     OF     ELECTRIC-CURRENT      SUPPLY     TO     ROTARY 

CONVERTERS 177 

Fundamental  Diagram — Fundamental  Equation — Application  of  the 
Diagram.  Representation  of  Converter  Operation  with  Constant 
Potential  at  Primary  Terminals  and  at  Brushes — General  Case.  Reactive 
Current  Values  for  a  Given  Voltage  Variation  as  a  Function  of  the 
Load — Most  Suitable  Value  of  Current-Supply  Voltage — Most  Suitable 
Value  of  Reactance — Regulation  of  Voltage  at  Terminals  by  Variation 
of  the  Supply  E.M.F. — Regulation  of  Voltage  at  Terminals  by  Variation 
of  Reactance — Power-Factor  of  the  Generator. 


CHAPTER  IT. — PREDETERMINATION  OF  THE  FIELD — EXCITATION 

OF  ROTARY  CONVERTERS 194 

Characteristic  Features  of  the  Rotary  Converter — Compound- 
Excitation.  Different  Factors  of  this  Excitation — Determination  of 
Reactive  Current  as  a  Function  of  the  Excitation  when  the  Active  Current 
is  Constant,  then  when  the  Power  is  Constant,  the  Generator  E.M.F. 
being  always  Constant — Different  Values  of  the  Excitation,  with  Con- 
stant Power  and  Constant  Potential.  V-Curves  for  Constant  Potential — 
Upper  Limit  of  Reactive  Current — Lag-Characteristics  of  Rotary  Con- 
verters at  Constant  Potential — Effective  Characteristic  of  Rotary  Con- 
verter under  Load — Application  in  the  Case  of  Separate  Excitation — 
Application  to  the  General  Case  of  Self-Excitation — Regulation  of 
Supply  E.M.F.  by  Compounding  of  the  Generator — Regulation  of 
Voltage  by  Varying  the  Reactance  X  in  the  Circuit — Possibility  of 
Suppressing  the  Shunt-Winding — Conclusion. 


CHAPTER  III. — STABILITY    OF    OPERATION    OF    ROTARY    CON- 
VERTERS    213 


CHAPTER  IV. — OPERATION  OF  SEVERAL  ROTARY  CONVERTERS  IN 

PARALLEL 219 

Inherent  Oscillations  or  Pumping  of  Converters  Connected  in  Parallel — 
Use  of  Rotary  Converters  for  Transforming  Direct  into  Alternating 
Current — Other  Special  Applications  of  Converters — Phase-Converters. 


CHAPTER  V.— VOLTAGE  RATIO  IN  SYNCHRONOUS  CONVERTERS 
WITH  SPECIAL  REFERENCE  TO  THE  SPLIT-POLE  CON- 
VERTER   225 


CONTENTS  xiii 

PART  HI 

PAGE 

CHAPTER  I. — METHODS  OF  CALCULATION  OF  THE  ARMATURE 
REACTIONS  (DIRECT  AND  TRANSVERSE)  OF  ALTER- 
NATORS   236 

Principles  of  the  Theory  of  Two  Reactions — Diagram  of  E.M.F.'s  and 
Current  of  an  Alternator  with  Unsaturated  Armature  and  with  Saturated 
Field  Magnet — Diagram  of  Ampere-Turns  in  the  Case  of  Unsaturated 
Armature — Remark  No.  i,  Upon  the  Case  of  an  Unsaturated  Arma- 
ture— Remark  No.  2,  Upon  the  Subject  of  Diagram  No.- 1 — The  Case 
of  a  Saturated  Armature — Local  Corrections  of  the  Air-Gap  Due  to 
Saturation  (Second  Approximation) — Case  of  Field  Magnets  with 
Divided  Windings — Practical  Calculation  of  Reactions — Comparison 
with  Theoretical  Coefficients — Case  of  Single-Phase  Alternators — 
Consequences  from  the  Point  of  View  of  the  Construction  of  Alter- 
nators for  Good  Regulation. 

CHAPTER  II. — METHODS  OF  TESTING  ALTERNATORS  ACCORDING 

TO  THE  THEORY  OF  TWO  REACTIONS 270 

Method  No.  i.  When  the  Rigid  Coupling  of  the  Two  Alternators  is 
possible — Method  No.  2.  Applicable  to  a  Single  Synchronous  Machine 
Operating  upon  an  Actual  Conducting  System — Analogies  between  this 
Method  and  that  of  Potier  Behrend — Method  No.  3 — For  the  Determina- 
tion of  Transverse  Reaction  (Coefficient  L). 

APPENDIX  A 283 

APPENDIX  B , 285 

APPENDIX  C 287 

INDEX 291 


INTRODUCTION 


Classification  of  Singlephase  and  Polyphase  A.C.  Motors.     An 

alternating  current  motor  comprises,  like  any  dynamo-electric  machine, 
an  inducing  magnetic  field  and  its  induced  circuit,  the  one  turning  with 
respect  to  the  other.  But,  while  the  field  of  D.C.  motors  is  always 
constant,  that  of  A.C.  motors  may  be  either  constant,  alternating,  or 
revolving,  according  to  whether  it  be  produced  by  a  direct  current, 
an  alternating  current,  or  a  system  of  polyphase  alternating  currents 
serving  to  excite  windings  suitably  interlaced.  Therefore,  as  was 
proposed  quite  logically,  in  1891,  by  E.  Hospitalier,1  A.C.  singlephase 
and  polyphase  motors  can  be  classified  according  to  the  nature  of  their 
magnetic  fields,  into  three  classes: 

(1)  Constant  field  motors. 

(2)  Alternating  field  motors. 

(3)  Revolving  field  motors. 

The  first  class  constitutes  the  subject-matter  of  the  present  book, 
the  second  and  third  classes  being  reserved  for  another  volume.  As 
will  be  seen,  these  three  classes  each  contain  singlephase  and  poly- 
phase motors. 

Constant  field  motors  can  also  be  characterized  by  the  fact  that  the 
armature-rotation  can  be  maintained  at  a  single  speed  only,  which  is 
synchronous  with  the  alternations  of  the  currents  employed.  We 
will  therefore  call  them,  more  frequently,  in  accordance  with  common 
custom,  synchronous  motors,  in  contradistinction  to  the  two  other 
classes,  which  may  be  characterized  as  asynchronous  or  non-synchronous. 
For  the  sake  of  greater  precision,  we  will  apply  the  latter  qualification 
to  motors  of  the  last  class  only;  and  since  alternating-field  motors 
(with  one  single  exception,  which  is  of  little  importance)  are  char- 
acterized by  the  use  of  a  commutator  similar  to  that  used  with  D.C. 

1  Bulletin  de  la  Societe  francaise  de  physique,  17.  Juillet,  1891. 

XV 


xvi  INTRODUCTION 

machines,  we  will  give  them  the  more  distinctive  name  of  Commutator 
Motors. 

Synchronous  Motors.  A  synchronous  motor  can  be  denned  as 
being,  merely,  an  alternator  used  as  a  motor.  The  transmission  of  power 
between  an  A.C.  generator  and  an  A.C.  motor  is,  therefore,  nothing 
more  than  a  particular  case  of  the  coupling  of  two  alternators  in  syn- 
chronous operation.  Indeed,  it  is  precisely  through  the  study  of  the 
features  of  the  coupling  of  alternators  in  parallel  that  the  occasion 
presented  itself  of  noting  the  phenomenon  of  the  reversibility  of  alter- 
nators, that  is  to  say,  the  possibility  of  using  the  same  machine  both 
as  a  motor  and  a  generator,  provided  that  it  shall  have  been  previously 
brought  to  a  speed  absolutely  equal  to  that  of  the  generator  which 
supplies  it  with  current. 

We  can  easily  understand  the  possibility  of  operating  such  a  motor 
by  comparing  it  to  a  motor  with  commutated  current.  It  is  known 
that  if  the  current  of  a  shuttle  armature  of  the  Siemens  ("H' ')  type 
is  commutated  at  each  half  revolution,  the  motor-couple  is  always 
in  the  same  direction  when  the  machine  is  supplied  by  direct  current. 
In  an  A.C.  system,  the  same  result  is  obtained  without  a  commutator, 
because  the  direction  of  the  supply-current  changes  at  each  half  revolu- 
tion, and  this  effect  occurs  only  when  the  motion  of  the  motor  is  syn- 
chronous, that  is  to  say  when  the  armature  advances  the  distance  of 
one  pole  during  one  alternation  of  the  supply-current. 

Although  this  property  was  noted  as  early  as  1869  by  Wilde,1 
it  passed  unnoticed  during  more  than  ten  years,  and  it  has  really  been 
known  only  since  the  experiments  of  J.  Hopkinson  and  Grylls- Adams, 
at  the  South  Foreland  Lighthouse,  in  1883.  The  Memoir  of  Hopkin- 
son2 (in  which,  without  knowing  the  work  of  Wilde,  he  gives  the 
explanation  to  which  reference  will  be  made  later) ,  was  epoch-making 
in  the  history  of  alternating  currents. 

In  the  South  Foreland  experiments,  the  alternators  used  were  three 
similar  de  Meritens  singlephase  alternating  current  machines,  all 
belt-driven  from  a  common  source  of  power. 

1  Wilde,  On  a  Properly  of  the  Magneto-electric  Current  to  Control  and  Render 
Synchronous  the  Rotations  of  the  Armature  of  a  Number  of  Electromagnetic  Induction 
Machines,  Philosophical  Magazine,  January,  1869. 

2  Hopkinson,  On  the  Theory  of  Alternative  Currents,  Particularly  in  Reference  to 
Tiuo  Alternate  Current  Machines  Connected  to  the  Same  Circuit,   Journal  of  the 
Society  of  Telegraph  Engineers,  1884,  P-  496,  Vol.  XIII.     See  also  the  paper  on 
The  Alternate  Current  Machine  as  a  Motor,  by  Grylls- Adams,  presented  at  the 
same  meeting. 


INTRODUCTION  xvii 

These  machines  could  be  easily  coupled  in  parallel,  as  generators, 
by  bringing  them  to  the  same  speed  before  coupling  them.  The  belt 
being  then  removed  from  one  of  them,  it  was  observed  that  it  con- 
tinued to  run  synchronously  by  the  action  of  the  current  of  its  neighbors, 
and  that  it  could  even  develop  a  considerable  amount  of  power,  as 
measured  by  a  friction  brake,  before  losing  its  synchronism.  These 
experiments  were  repeated  a  few  years  later  by  Mordey,  on  a  much 
larger  scale,  with  machines  of  low  inductance  presenting  a  much  greater 
stability  of  operation  and  driven  by  independent  prime  movers.  He 
was  thus  able  to  demonstrate  the  synchronizing  power  of  the  alternators 
on  the  motors  or  engines  driving  them,  and  even  to  cause  one  of  the 
latter,  with  the  power  shut  off,  to  be  dragged  by  one  of  the  alternators 
which  it  was  driving.  This  gives  the  key  to  the  principles  involved 
in  parallel  working.  He  also  showed,  later,  the  possibility  of  accom- 
plishing this  coupling  with  machines  connected  by  means  of  long  lines 
of  high  resistance. 

Synchronous  singlephase  motors  have  two  great  disadvantages: 
they  are  not  self-exciting,  and  they  cannot  start  alone,  even  without 
load.  Zipernowsky  was  the  first  to  overcome  this  difficulty  by  the 
expedient  of  adding  a  commutator  to  his  motors,  which  enables  them 
to  be  started  with  alternating  current,  and,  after  they  have  attained 
synchronous  speed,  to  be  excited  by  a  portion  of  the  alternating  current 
which  they  consume. 

These  motors,  manufactured  by  the  firm  of  Ganz  &  Co.,  had  a 
certain  vogue,  in  consequence  of  the  tests4 made  of  them  at  Frank- 
fort, in  1899,  by  a  Technical  Commission.  The  efficiency  was  satis- 
factory, being  77  per  cent  for  motors  of  15  H.P.,  and  86  per  cent  for 
motors  of  30  H.P.1  This  system  is  no  longer  used  at  the  present  time, 
except  for  small  powers  (i  to  5  H.P.). 

When  the  invention  of  polyphase  currents  became  known,  it  led 
naturally  to  the  idea  of  utilizing  them  for  the  transmission  of  power 
between  two  synchronous  machines  of  the  same  type.  Bradley,  in 
America,  and  Haselwander,  in  Germany,  took  out  patents,  as  early 
as  1887,  the  former  on  a  two-phase  synchronous  motor,  and  the  latter 
on  a  three-phase  synchronous  motor.  In  both  cases  the  motor  was 
produced  by  making  taps  on  a  Gramme  ring  and  connecting  these 
with  insulated  rings  mounted  on  the  armature-shaft.  Non-synchronous 
motors  were  only  invented  in  the  following  year,  by  Ferraris  and  by 
Tesla. 

1  La  Lumiere  Electrique.  Vol.  XXXII  (1889),  p.  328. 


xviii  INTRODUCTION 

It  was  in  1891,  at  the  Frankfort  Exhibition,  that  synchronous  poly- 
phase motors,  with  flat  ring  or  Gramme  ring  armatures,  constructed 
by  the  firms  of  Schuckert  and  of  Lahmeyer,  and  of  sizes  as  large  as 
50  H.P.,  were  seen,  for  the  first  time,  alongside  the  first  non-synchronous 
motors  of  Dolivo-Dobrowolski  and  of  Brown. 

Since  that  time  the  principle  of  synchronous  motor  operation  has 
been  extended  to  ordinary  polyphase  alternators,  with  any  winding 
whatever,  stationary  or  movable,  with  poles  alternating  or  not;  and 
the  only  improvements  that  have  been  made  have  been  in  the  means 
of  their  excitation  or  of  starting  them. 

In  1890-1  Swinburne  had  had  the  idea  of  producing,  by  means  of 
over-excited  synchronous  motors,  the  relatively  considerable  magnetiz- 
ing current  consumed  by  his  "  hedgehog  "  transformers.  Under  these 
conditions  the  motor  played  the  same  role  as  a  condenser.  This  very 
interesting  property  was  utilized  industrially  in  1893  in  the  Bulach- 
Oerlikon  power-transmission  installation,  at  the  suggestion  of  Dolivo- 
Dobrowolski;  and  it  was  also  used  in  the  distributing  system  at  Bocken- 
heim,  by  Lahmeyer,  to  compensate  for  the  wattless  current  of  the 
non-synchronous  motors,  and  even  for  raising  the  voltage  of  the  gen- 
erators. This  method  has  come  into  extensive  use  at  the  present  time, 
especially  in  the  United  States. 

It  constitutes  an  advantage  in  favor  of  synchronous  motors,  and 
it  has  prevented  them  from  disappearing  from  the  commercial  field, 
after  the  development  of  motors  of  revolving  field  type,  which  are  quite 
superior  to  them  from  other  standpoints,  notably  in  regard  to  starting 
power.  It  is  desirable  to  utilize,  wherever  possible,  both  these  types 
of  motors  in  distributions  of  mechanical  power.  The  synchronous 
polyphase  motors  are  especially  useful.  They  start  readily  without 
load,  are  self -exciting,  and  have  an  efficiency  equal  to  that  of  alternators. 

The  principal  objection  to  synchronous  motors  (which  is,  however, 
in  certain  cases,  an  advantage)  is  the  impossibility  of  modifying  their 
speed  of  rotation  (without  modifying  that  of  the  generators).  Revolv- 
ing field  motors  are  superior  to  them  in  that  respect,  but  this  advantage 
is  obtained  at  the  expense  of  efficiency.  Commutator  motors,  of 
which  no  mention  will  be  made  here,  are  alone  capable  of  running 
at  all  speeds,  the  same  as  direct  current  motors;  and  for  this  reason 
they  present  a  certain  amount  of  interest. 


PART  I 

SYNCHRONOUS  MOTORS 


CHAPTER  I 


GENERAL    PRINCIPLES    OF    SYNCHRONOUS    MOTORS 

Construction.  Synchronous  motors  have  the  same  construction 
as  alternators.  The  few  special  features  relative  to  the  production 
of  the  direct  current  necessary  for  their  excitation  will  be  treated  sepa- 
rately, later.  It  will  be  assumed 
that  the  reader  is  already  familiar 
with  the  general  details  of  con- 
struction of  alternators. 

There  are  motors  having  mov- 
able armatures  and  stationary  fields, 
or  vice  versa,  and  also  motors  with 
revolving  iron  masses  in  which  all 
the  windings  are  stationary.  These 
machines  are  similar  to  the  gen- 
erators of  the  same  types;  for 
example,  Fig.  i  indicates,  diagram- 
matically,  the  principle  of  construc- 
tion of  a  two-phase  synchronous  motor,  with  a  ring  armature  and  mov- 
able fields,  receiving  an  exciting  current  through  the  brushes  bi  and  b2. 

These  motors  are  designed  like  generators,  the  essential  condition 
to  be  fulfilled  being  to  have  a  low  armature-reaction  and  powerful 
inducing  fields,  in  order  to  obtain  good  stability. 

Although  it  is  more  difficult  to  increase  the  number  of  poles 
for  small  powers  than  for  large  powers,  the  construction  of  small 


FIG.  i. 


SYNCHRONOUS  MOTORS 


synchronous  motors  for  ordinary  frequencies  (40  to  60  cycles) 
presents  no  special  difficulties,  if  the  speeds  corresponding  to  these 
frequencies  are  not  objectionable,  because  these  speeds  are  perfectly 
allowable  so  far  as  centrifugal  force  is  concerned. 


On  the  other  hand,  in  the  construction  of  small  synchronous  motors 
to  run  at  low  angular  velocities,  it  is  extremely  difficult  to  find  space  for 
the  numerous  conductors  and  for  the  exciting  or  field  coils,  which 


GENERAL  PRINCIPLES  OF  SYNCHRONOUS  MOTORS       3 

must  produce  as  many  ampere-turns  as  in  the  case  of  large  motors. 
For  this  reason  non-synchronous  motors  are  more  convenient  for  low 
rotative  speeds. 

The  author  has  been  able,  however,  to  produce  motors  of  low  power 
(a  few  hundred  watts)  which  have  moving  iron  and  have  a  very  high 
number  of  poles  (as  many  as  50  for  example),  by  utilizing  induction- 
type  excitation,  the  magnetic  circuit  being  closed  exteriorly,  as  shown  in 
Fig.  2,  in  such  a  way  as  to  allow  all  the  space  needed  for  the  exciting 
coils. 

These  coils  can  then  be  replaced  by  permanent  magnets,  thus 
producing  motors  which  run  without  excitation,  at  speeds  sufficiently 


FIG.  3. 

low  to  be  synchronized  by  hand,  and  which  can  render  useful  service, 
in  certain  applications,  such  as  for  oscillographs.  For  this  purpose 
the  author  preferably  employs  a  small  horseshoe  magnet  that  is  made 
to  revolve  around  a  stationary  armature  having  a  number  of  poles 
which  is  a  multiple  of  6.  It  is  possible,  in  this  way,  to  obtain  very 
stable  synchronous  rotation  of  a  revolving  mirror  without  expending 
more  than  ij  to  2  watts. 

Several  firms  made  a  specialty  of  synchronous  motors,  at  an  early 
date,  among  which  we  may  mention  La  Societe  1'Eclairage  Electrique 
in  France,  and  the  Fort  Wayne  Company  in  America. 

One  form  of  motor  constructed  in  France  by  the  Societe  1'Eclairage 
Electrique  (Figs.  3  and  4),  is  constructed  for  polyphase  currents  or 
for  single-phase  currents,  for  powers  ranging  from  i  to  130  H.P.  The 
table  on  page  5  gives  the  principal  data  referring  to  these  matters. 


4  SYNCHRONOUS  MOTORS 

The  efficiencies  of  the  three-phase  motors  are  a  little  higher  than 
those  given  for  the  single-phase  motors.  The  horse-powers  given  in 
this  table  correspond  to  a  frequency  of  42  periods,  but  these  motors 
can  be  also  used  at  frequencies  between  40  and  60  periods,  and  their 
power  then  increases  with  the  frequency. 

As  the  table  shows,  types  Nos.  14  to  30  are  made  with  4  poles, 
self-exciting.  For  higher  powers,  the  number  of  poles  increases,  and 
the  excitation  is  obtained  by  means  of  a  small  direct  current  exciter 
mounted  on  the  same  base.  Above  type  90  the  armature  is  stationary 
and  the  fields  turn  inside.  The  fields  are  of  mild  cast  steel,  the  arma- 
tures being  slotted. 


FIG.  4. 

As  an  example  of  these  large  motors  may  be  cited  several  from  50 
to  100  H.P.,  giving  the  best  of  results  on  the  power-transmission  system 
around  Grenoble,  notably  at  Voiron,  a  distance  of  30  kilometers  from 
the  generating  station.  Their  efficiency  is  from  90  to  92  per  cent. 
One  of  these  motors  even  works  in  parallel  with  a  steam-engine  of  the 
same  power,  and  it  compensates  for  the  variation  of  angular  velocity 
of  the  engine  as  it  passes  the  dead  centers. 

All  these  motors  are  provided  with  a  clutch  and  with  an  idle  pulley 
for  starting,  as  will  be  explained  later.  When  running,  they  can 
undergo  considerable  variations  of  load  without  falling  out  of  step. 

Attention  should  also  be  called  to  another  interesting  type  of  syn- 
chronous motor,  the  Maurice  Leblanc  type,  which  is  characterized 
by  the  addition  of  closed  circuits  in  the  pole-pieces  to  insure  a  perfect 
damping  of  oscillations,  as  will  be  seen  later. 


GENERAL  PRINCIPLES  OF  SYNCHRONOUS  MOTORS 


SINGLE-PHASE  AND   THREE-PHASE  SYNCHRONOUS   MOTORS   OF 
THE  "SOCIETE  L'ECLAIRAGE  ELECTRIQUE" 


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Experimental  Properties.  As  already  stated,  it  is  an  experimental 
fact  that  synchronous  motors  can  only  be  run  after  they  have  first  been 
brought  to  synchronous  speed  by  some  external  means.  As  will  be 
seen  later  the  motors  themselves  can  run  indifferently  in  either  direc- 
tion, but  the  direction  of  rotation  selected  in  bringing  them  to  syn- 
chronism should  be  that  which  is  suitable  for  the  brushes  of  the  motor 
or  of  the  exciter. 

Case  of  Equal  Electromotive  Forces.  Let  us  suppose  that  the 
electromotive  forces  of  the  generator  and  motor  are  equal,  and,  to 
simplify  matters,  let  us  take,  as  generator  and  motor,  two  machines 
whose  excitations  are  regulated  to  approximately  the  same  value. 
Let  the  two  machines  be  driven  by  belts  (Fig.  5) ;  and,  when  they  have 
attained  the  same  speed,  let  them  be  coupled  together  (experiment 
of  Hopkinson  and  Grylls-Adams) .  Let  us,  moreover,  make  use  of  an 
apparatus  of  the  kind  described  in  Chapter  VII,  whereby  the  difference 
of  phase  between  the  two  machines  may  be  determained. 

It  will  be  noted,  in  the  first  place,  that  as  soon  as  the  two  machines 
are  brought  to  the  same  speed,  the  current  which  passes  from  the  one 


6 


SYNCHRONOUS  MOTORS 


to  the  other  practically  disappears.  Moreover,  the  "  phases  are  iden- 
tical," i.e.,  the  poles  of  like  polarity  pass  at  the  same  time  in  front  of 
the  corresponding  portions  of  the  two  armatures. 

The  induced  E.M.F.'s  between  the  corresponding  terminals  a, 
b,  and  A,  B,  are  therefore  in  unison.  If  we  measure  them,  on  the  con- 
trary, in  the  direction  in  which  they  appear,  by  following  the  circuit 
ab,  BA,  it  will  be  found  that  they  are  exactly  opposed  to  each  other. 

Let  us  now  suppose  the  belt  of  one  of  the  two  machines  to  be  removed. 
This  machine  will  continue  to  turn  at  the  same  speed,  but  it  gives  indi- 
cation of  a  certain  very  slight  delay  or  falling  behind,  technically 
termed  "lag,"  with  respect  to  the  other  machine.  Moreover,  the 
current  in  the  circuit  now  becomes  appreciable. 

If  a  brake  be  placed  on  the  pulley  and  if  the  load  be  gradually 
increased  in  such  a  way  as  to  increase  the  mechanical  power  produced 


FIG.  5. 


by  the  motor,  the  "  lag  "  of  the  motor  will  be  seen  to  increase  at  the 
same  time  as  the  current. 

When  this  lag  approaches  a  quarter  of  a  period,  i.e.,  half  an  inter- 
polar  space,  the  machine  slows  up  all  at  once  and  stops  as  if  held  fast 
by  the  brake.  We  then  say  that  it  is  "  stalled,"  or  "  out  of  synchron- 
ism," or  "  out  of  step."  The  current  in  the  circuit  rises  to  a  very  high 
value  as  soon  as  the  machine  falls  out  of  synchronism;  and  it  becomes 
approximately  equal  to  the  short-circuit  current  in  the  circuit  when  the 
machine  is  stopped.  In  order  to  avoid  accidents,  it  is  necessary  to 
introduce  fuses  in  the  circuit,  or  to  provide  some  automatic  disconnecting 
device,  which  will  prevent  the  excessive  load. 

It  is  seen  that  what  characterizes  the  synchronous  motor  is  the 
increase  of  phase -lag  with  the  load  and  the  "  stalling  "  of  the  motor 
or  its  falling  out  of  step  beyond  a  certain  maximum  load. 

In  a  good  motor,  the  limiting  load  should  amount  to  at  least  1.5 
times,  or,  better,  to  twice  the  normal  load.  This  limit  is  guaranteed 
by  most  makers  of  synchronous  motors. 

On  the  other  hand,  if  the  motor  is  run  by  a  belt  in  such  a  way  as  to 
give  it  a  "  lead  in  phase,"  with  respect  to  the  machine  or  the  circuit 


GENERAL  PRINCIPLES  OF  SYNCHRONOUS  MOTORS       7 

which  supplies  it  with  current,  it  can  be  found,  by  wattmeter  measure- 
ments, that  this  power  changes  in  sign,  i.e.,  the  motor  acts  as  a  brake 
and  returns  energy  to  the  circuit  instead  of  receiving  it  therefrom. 

The  phenomena  become  more  complicated  still  on  varying  the 
E.M.F.  of  the  motor  or  of  the  generator. 

Case  of  Unequal  Electromotive  Forces.  An  interesting  and  char- 
acteristic property  of  synchronous  alternating  current  motors,  and 
which  distinguishes  them  absolutely  from  direct  current  motors  or  from 
alternating  current  motors  having  commutators,  is  that  they  can  be 
excited  so  as  to  give  a  voltage  greater  than  that  of  the  supply-circuit.  For 
example,  it  is  possible  to  feed,  from  a  no- volt  circuit,  a  motor  which, 
driven  by  belt  at  the  same  speed,  produces  an  E.M.F.  of  120  to  150  volts 
at  its  terminals.  But,  if  the  E.M.F.'s  are  thus  unequal,  the  current  pass- 
ing between  the  generator  and  the  motor,  when  the  latter  is  running  with- 
out load,  can,  instead  of  being  inappreciable,  attain  a  considerable  value. 
Likewise,  when  the  motor  is  running  with  load,  the  current  is  greater 
than  that  which  corresponds  to  the  work  to  be  done.  The  same  effects 
are  produced  when,  instead  of  giving  to  the  motor  an  excessive  excita- 
tion, it  is  given  an  insufficient  induced  E.M.F.  It  is  then  observed,  if 
the  machines  are  alike,  that  the  potential  difference  at  the  terminals 
assumes  a  third  value,  which  is  the  mean  of  the  two  E.M.F.'s  involved. 

In  both  cases,  the  greater  the  inequality  between  the  two  E.M.F.'s 
the  more  the  current  measured  will  increase,  by  the  change  of  excita- 
tion. If  we  plot  a  diagram,  taking,  as  abscissae,  the  values  of  the  excita- 
tion of  one  of  the  machines,  and,  as  ordinates,  the  current  passing 
through  the  circuit,  the  curve  of  variation  of  the  latter,  as  a  function 
of  the  former,  has  the  form  of  a  V  more  or  less  rounded  at  the  bottom 
(Fig.  6).  This  form  persists,  although  it  may  be  less  marked,  when 
a  constant  load  is  placed  on  the  brake.  At  the  same  time  that  the 
current  increases,  by  reason  of  an  inequality  of  the  E.M.F.'s,  it  can  be 
noted,  by  means  of  an  apparatus  for  indicating  phase-difference,  that 
the  current  undergoes  a  change  of  phase,  either  forward  or  backward, 
with  respect  to  the  E.M.F.  of  the  motor.  This  can  be  expressed  in 
another  way  by  saying  that  the  machine  consumes  or  produces  wattless 

current,1  i.e.,  current  which  is  "out  of  phase,"  being  -  behind  or  ahead 

of  the  E.M.F.  This  "  wattless  "  current,  which  has  the  effect  of  increas- 
ing the  "apparent"  current,  is  thus  named  because  it  produces  no 
work,  the  load  on  the  brake  remaining  constant,  by  hypothesis. 

1See  note  at  bottom  of  page  42. 


8 


SYNCHRONOUS  MOTORS 


The  effect  of  this  wattless  current  is,  therefore,  to  produce,  in  the 
motor,  a  supplemental  positive  or  negative  E.M.F.,  which  adds  itself 
to  its  own  induced  E.M.F.,  in  such  a  way  as  to  produce,  at  the  terminals, 
a  difference  of  potential  equal  to  that  of  the  generator.  We  can  con- 
clude from  this,  without  further  argument,  that  when  the  motor  gen- 
erates an  E.M.F.  which  is  too  low,  the  current  of  the  generator  tends 
to  over-excite  it  and  that,  in  the  contrary  case,  it  tends  to  under-excite  it. 

The  action  of  the  current  on  the  generator  itself  produces  inverse 
effects. 

'    The  effects  are  more  complicated  still  when  resistances  or  induct- 
ances are  added  in  the  circuit  between  the  machines,  with  the  general 


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t 

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1  \ 

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f 

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V 

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9       ?        4 

^^.8         /O      /Z^    /4       /6         i&        25 
Excirarion  Currenr  Amperes 

FIG.  6. 

effect  of  lowering  the  voltage.     Synchronous  operation  remains  possible, 
nevertheless,  even  when  the  resistance  attains  high  values.1 

When  the  circuit  includes  reactance  it  is  observed  that,  by  over- 
exciting  the  motor,  the  voltage  will  be  raised  at  its  terminals,  and  even 
at  the  terminals  of  the  generator,  so  as  to  attain  values  which  are  higher 
than  the  E.M.F.  of  the  generator,  measured  with  open  circuit.  On  the 
contrary,  by  under-exciting  the  motor,  it  is  possible  to  produce  increas- 
ing and  rapidly  exaggerated  voltage-drop  along  the  whole  line. 

1  When  the  resistance  is  high,  say  75  per  cent,  it  may  be  necessary  to 
add  some  external  reactance,  so  as  to  cause  the  current  to  lag  more  behind 
the  cross  E.M.F.— (E.  J.  B.) 


GENERAL  PRINCIPLES  OF  SYNCHRONOUS  MOTORS       9 

An  over-excited  synchronous  motor,  connected  to  the  terminals  of  an 
alternator  having  excessive  armature-reaction,  can  even  replace  the 
excitation  of  the  latter.  It  is  observed,  indeed,  that  on  suppressing 
this  excitation,  the  generator  continues  to  run  the  motor,  and  furnishes 
the  normal  voltage  at  its  terminals;  but  it  can  develop  only  little  power. 
An  over-excited  motor  thus  produces  an  indirect  self -excitation  which  is 
equivalent  to  that  obtainable  from  a  condenser.  There  is,  in  other  re- 
spects, a  complete  analogy  of  effects  between  the  two  forms  of  apparatus. 

These  experimental  results  are  much  too  complex  to  be  studied 
more  in  detail  here.  They  can  be  discussed  more  satisfactorily  later, 
in  connection  with  the  theory  of  these  motors  and  their  applications. 

Elementary  Explanation  of  Polyphase  Synchronous  Motors.  If 
we  turn  our  attention,  first,  to  polyphase  synchronous  motors,  the 
explanation  of  the  phenomena  just  described  is  made  easy  by  the  con- 
sideration of  revolving  magnetic  fields. 

For  the  sake  of  brevity  we  will  adopt  the  terms  "rotor  "  and  "  stator  " 
to  designate  the  movable  and  fixed  portions,  respectively,  of  the  motor, 
in  accordance  with  the  terminology  of  Professor  S.  P.  Thompson. 

Let  us  take,  as  an  example,  a  motor  having  two  pairs  of  pole-pieces, 
in  which  the  inductive  circuit  is  of  movable  form  (rotor)  and  the 
induced  is  circuit  of  stationary  form  (stator). 

In  the  ordinary  form  of  polyphase  alternators,  the  rotor  will 
consist  of  a  crown  of  iron  cores,  with  protruding  poles  excited  by 
coils  receiving  direct  current  from  a  separate  exciter.  The  stator, 
on  the  other  hand,  will  consist  of  a  circular  core  of  laminated  iron 
having  some  induced  windings  disposed  in  notches  or  slots  in  such 
a  manner  that  the  wires  in  the  successive  slots  shall  have  alternating 
currents  of  different  phase  passing  through  them.  If  we  suppose, 
for  example,  that  we  have  a  winding  for  four  poles  and  for  six  phases 
(three  slots  per  pole)  such  as  is  shown  in  Fig.  7,  the  wires  in  the  six 
slots  which  cover  two  poles  of  the  stator,  as  we  follow  along  the  periphery 
of  the  latter,  will  have,  passing  through  them,  six  currents  l  which  are 
out  of  phase  with  respect  to  each  other  by  %  of  a  period,  and  which  can 
be  represented  by  the  equations 

i i  =  /o  sin  a>t ; 


sin     tut 


22LV 
6/' 


1  In  reality,  the  six  phases  are  supplied  by  three-phase  currents  only;  the 
windings  which  are  of  exactly  opposite  phases  being  connected  in  series,  with 
reversed  connections. 


10 


SYNCHRONOUS  MOTORS 


?3=/o  sin  (cot—  2~^\> 


=      sn    ^~ 


=      sn    w*- 


in  which 


27T\ 

-5—  I; 


A* 


FIG.  7. 

T1  being  the  common  duration  of  the  period  of  the  alternating  currents 
considered,  IQ  their  common  amplitude  (i.e.,  maximum  value),  and 
i\,  *2,  *s,  *4,  *5i  *e,  being  the  currents  in  the  slots  1,2,3,  4,  5,  6. 

It  will  be  seen  that,  at  every  one-sixth  of  a  period,  the  currents  in 
the  stator  resume  the  same  values,  but  the  latter  are  displaced  one- 
sixth  of  the  width  of  a  double  field  (2  poles)  in  the  direction  in  which 
the  currents  succeed  each  other  along  the  stator.  Therefore,  the  axes 
radiating  from  the  magnetic  fields  produced  by  the  windings  of  the 
stator  displace  themselves  around  this  stator  with  an  angular  velocity  ; 


__ 

- 


27T 


corresponding  to  a  number  of  turns  =  —  =  per  minute. 


GENERAL  PRINCIPLES  OF  SYNCHRONOUS  MOTORS      11 


The  magnetic  strength  of  these  fields  can  be  considered  practically 
constant  inasmuch  as  it  also  resumes  the  same  value  at  every  one-sixth 
of  a  period  (although  it  may,  in  the  intervals,  undergo  slight  variations 
which  are  dampened  by  the  hysteresis  and  the  eddy  currents 
produced  in  the  pole-pieces  of  the  rotor).  Therefore,  even  though  the 
armature  (stator)  may  be  stationary,  the  result  is  the  same  as  if  it  had 
rervolving  poles  which  attract  or  repel  the  poles  of  the  field  (rotor) 
and  we  can,  henceforth,  reason  as  if  we  were  dealing  with  the  attrac- 
tions of  two  systems  of  magnets  presenting  the  same  number  of  poles 
which  are  alternately  north  and  south  in  polarity.  (Fig.  8.)  The 
poles  of  unlike  polarities,  of  the  two  sys- 
tems, attract  each  other;  the  others  repel 
each  other.  Therefore,  when  at  rest,  the 
poles  of  unlike  polarity  will  always  face 
each  other.  If  the  external  magnets  begin 
to  rotate  slowly,  starting  from  rest,  they  will 
drag  with  them  the  stationary  magnets, 
whose  poles  tend  to  remain  opposite  the 
poles  of  unlike  polarity.  (This  result  may 
be  obtained  by  supplying  the  motor  with 
current  obtained  from  a  generator  which 

is  started  from  rest  and,   consequently,  gives    polyphase   currents   of 
increasing  frequency.) 

The  attractions  can  only  be  concordant  and  continuous  when  the 
two  systems  turn  at  the  same  speed;  which  explains  the  necessity  of 
synchronism.  Otherwise,  there  would  only  be  successive  attractions 
and  repulsions  which  would  neutralize  each  other. 

The  stability  of  synchronous  operation  is  also  easily  demonstrated 
by  considering  the  moment  of  the  motor-couple  (i.e.,  the  torque).  If 
the  poles  of  the  rotor  remain  opposite  the  revolving  poles  of  the  stator, 
the  attractions  produced  are  directed  radially  and  consequently  they 
produce  a  motor-couple  or  torque  which  is  equal  to  zero.  If,  on  the 
contrary,  for  any  reason  whatever,  the  rotor  loses  or  gains  speed,  some 
tangential  attractions  or  repulsions  will  appear,  whose  resultants  tend 
to  bring  back  the  opposite  poles  of  the  rotor  into  coincidence  with  the 
poles  of  unlike  polarity  of  the  stator,  so  long  as  the  poles  of  the  rotor 
remain  near  these,  because  the  attractions  of  unlike  poles  and  the 
repulsions  of  like  poles  act  in  the  same  direction;  but  if  the  difference 
in  phase  amounts  to  one  interpolar  space,  the  poles  of  like  sign  of  the 
rotor  and  stator  will  be  opposite  each  other,  the  motor-couple  or  torque 


FIG.  8. 


12 


SYNCHRONOUS  MOTORS 


will  become  zero,  and  will  then  change  sign  if  the  difference  in  phase 
increases.  By  reason  of  the  symmetrical  construction  of  the  motor 
the  torque  will  have  points  of  maximum  and  minimum  value  at  equal 
distances  between  the  points  of  zero-value,  i.e.,  in  the  positions  where 
the  poles  of  the  rotor  are  midway  between  the  poles  of  the  stator. 

To  sum  up,  taking  as  abscissae  the  difference  of  phase  /  of  the  poles 
of  the  rotor,  expressed  in  terms  of  the  interpolar  space  L,  and  taking 
as  ordinates  the  torque  C,  the  representative  curve  will  take  the  form 
shown  herewith  (Fig.  9),  the  magnetic  strength  at  the  armature-poles 
being  supposed  constant,  i.e.,  assuming  the  currents  that  produce  this 
magnetic  flux  to  be  constant. 

The  -machine  will  have  stable  operation  for  the  difference  of  phase 
comprised  between  the  two  maximum  points  B  and  C  (the  positive 


Operating  as  Motor 


,r          » 

*OperrTt-/ng as  *i  Gene  ra  for 

FIG.  9. 

maximum  being  due  to  a  lag,  and  the  negative  maximum  being  due 
to  a  lead),  because  every  accidental  advance  (or  lag)  is  corrected 
of  itself  by  a  contrary  variation  of  the  torque.  If  the  rotor  lags,  for 
example,  in  consequence  of  a  passive  mechanical  resistance,  the  increase 
in  torque  compensates  for  this  resistance. 

When  the  motor  is  running  without  load,  its  condition  corresponds 
to  the  position  O,  at  which  there  is  no  phase-difference.  When  the 
motor  is  loaded,  i.e.,  whenever  mechanical  resistance  is  applied  to  the 
shaft,  the  position  of  the  poles  of  the  rotor  changes  in  phase  and  comes 
to  a  point  O',  such  that  the  couple  O'm  shall  balance  the  resisting  couple. 
If  the  resisting  couple  is  greater  than  the  maximum  torque,  the  machine 
can  no  longer  run;  and  even  for  positions  of  m  which  are  a  little 
below  M,  the  machine  will  fall  out  of  step,  in  consequence  of  unavoid- 
able oscillations. 


GENERAL  PRINCIPLES  OF  SYNCHRONOUS  MOTORS     13 

On  the  other  hand,  to.  obtain  a  difference  of  phase  ahead,  between 
O  and  C,  it  is  necessary  to  apply  to  the  shaft  an  effort  in  the  direction 
of  rotation,  i.e.,  it  is  necessary  to  apply  to  the  shaft  a  certain  amount 
of  propelling  power  which  must,  evidently,  be  transformed  into  elec- 
trical energy. 

The  armature  current  has  been  supposed  constant.  In  practice, 
it  is  the  voltage  of  the  supply-circuit  which  is  constant,  at  its  terminals; 
and  the  question  is  thus  complicated  by  the  spontateous  variation  of 
the  current  with  the  variation  in  phase -difference.  This  variation, 
itself,  depends  on  the  ratio  of  the  induced  E.M.F.  of  the  motor  to  the 
voltage  applied  at  its  terminals. 

In  fact,  as  they  displace  themselves  before  the  armature  at  the 
synchronous  speed,  the  poles  of  the  inducing  field  induce  in  the  wind- 
ings counter  E.M.F.'s.  which  are  of  the  same  order  and  magnitude  as 
the  voltage  at  the  terminals.  If,  mentally,  we  locate  the  E.M.F.'s.  in  the 
wires  which  are  placed  in  the  slots,  we  perceive  readily  that  the  E.M.F. 
in  each  slot  varies  periodically  and  passes  through  a  maximum  at  the 
moment  when  the  middle  of  a  revolving  pole  comes  in  line  with  the 
slot.  The  axes  of  maximum  values  of  the  induced  E.M.F.'s  therefore 
coincide  with  the  axes  of  the  inducing  poles,  and  revolve  with  them. 
If  the  currents  were  in  phase  with  the  E.M.F.'s,  they  would  give  rise  to 
revolving  fields  whose  axes  would  be  retarded  in  phase  by  an  amount 
equal  to  half  the  width  of  a  pole,  since  each  conductor  forms  a  coil 
with  a  conductor  similarly  placed,  but  in  the  contrary  direction,  under 
the  next  pole. 

But  we  must  take  into  account  the  voltage  at  the  terminals,  with 
which  the  induced  E.M.F.  combines,  and  also  the  self-induction  of 
the  machine,  which  throws  the  current  out  of  phase  by  a  quarter  of  a 
period,  i.e.,  half  an  interpolar  space.  Therefore  the  question  can  only 
be  treated  with  precision  by  calculation,  as  will  be  seen  later.  From 
.the  qualitative  point  of  view,  the  result  differs  but  slightly  from  the 
preceding  result.  The  form  of  the  curve  of  torque  remains  analogous 
to  that  of  Fig.  9,  but  it  is  no  longer  so  symmetrical,  and  the  lags  OC 
and  OB,  which  determine  the  limits  of  stability,  take  a  value,  y,  which 

is  a  little  lower  than  — ,  and  which  is  defined  by  the  relation 

2 

a)L 
tan  r-  —  ; 

ajL  and  R  being,  respectively,  the  reactance  and  the  resistance  of  the 
armature  circuit. 


14  SYNCHRONOUS  MOTORS 

It  will  be  observed  that,  when  the  current  is  in  phase  with  the  induced 
E.M.F.,  the  magneto-motive  force  of  the  armature-reaction  has  no 
action  on  the  inducing  field,  and  can  only  produce  a  transverse  dis- 
tortion of  the  field;  while,  on  the  other  hand,  when  the  current  is  out 
of  phase  one-fourth  of  a  period  in  advance,  or  behind,  the  M.M.F.  is 
directly  opposed  to,  or  coincident  with,  that  of  the  field.  With  regard 

to  the  sign,  it  can  be  easily  seen  that  a  phase-difference  of  —  in  advance 
of  the  induced  E.M.F.  produces  a  magnetizing  reaction  which  is  the 
same  as  in  a  generator,  and  that  a  phase-difference  of  -  behind  pro- 
duces a  demagnetizing  reaction.  It  must  not  be  forgotten,  however, 
that  the  internal  E.M.F.  is  opposed  to  the  external  E.M.F.  and  that 
the  lag  and  lead  are  therefore  transposed  if  they  are  referred  to  the 
latter. 

Elementary  Explanation  of  Single-Phase  Synchronous  Motors. 
The  phenomena  are  more  complicated  in  single-phase  motors.  The 
same  explanation  may  nevertheless  be  retained  by  means  of  a  simple 
artifice  of  reasoning. 

The  coils  of  the  armature-winding,  being  excited  by  a  single  alternat- 
ing current  (Fig.  10),  produce  poles  which  no  longer  revolve,  but  are 
stationary.     These  poles  are  alternately 
positive  and  negative,  and  have  a  mag-  -*'lrtfl    8  lndut 

netic  flux  which  varies  periodically  like 
the  current  that  produces  it.  There  is, 
therefore,  no  tendency  to  rotation;  and 
the  motor  can  only  be  put  in  operation 
by  external  means,  as  already  seen.  But 
we  may  suppose  it  brought  previously 
to  synchronism. 

M.  Maurice  Leblanc  has  enunciated  a  theorem  which  is  an  elec- 
trical analogue  of  the  following  well-known  optical  theorem:  A 
vibration  of  luminiferous  ether  polarized  rectilinearly  may  be  replaced 
by  two  circularly  polarized  vibrations  of  contrary  sign  having  the  same 
frequency  and  having  amplitudes  equal  to  the  half  amplitude  of  the 
rectilinear  vibration. 

According  to  M.  Leblanc's  theorem,  an  alternating  stationary  mag- 
netic field  may  be  replaced  by  two  fields  revolving  in  contrary 
directions,  each  having  a  flux  half  as  large,  and  having  equal  velocities, 
such  that  they  advance  a  distance  equal  to  that  of  two  poles  during  a 


GENERAL  PRINCIPLES  OF  SYNCHRONOUS  MOTORS     15 

single  period.  The  fields  turning  in  the  same  direction  as  the 
inducing  poles  will  have  the  same  angular  velocity  a,  as  the  latter, 
and  will  drag  them  in  very  much  the  same  way  as  in  polyphase 
motors.  On  the  other  hand,  the  fields  which  turn  in  the  opposite 
direction  will  have  a  relative  velocity,  2  a,  which  is  contrary  to  and 
double  that  of  the  field-poles,  so  that  their  attracting  or  repelling 
actions,  since  they  succeed  each  other  in  inverse  directions,  will 
produce  no  resisting  torque.  These  reversed  revolving  fields  will 
give  rise  only  to  supplemental  losses  by  liysteresis  and  by  eddy 
currents. 

By  this  simple  analysis  (which  is,  in  reality,  only  approximate) 
the  operation  of  single -phase  motors  can,  it  is  seen,  be  discussed  and 
explained  in  the  same  way  as  that  of  polyphase  motors. 

It  has  been  supposed,  in  what  precedes,  that  the  armature  is  station- 
ary and  the  field  movable.  In  the  contrary  case  the  explanation  is 
the  same  if  we  consider  the  relative  velocities  of  the  two  portions,  but 
the  fields  displace  themselves  only  with  respect  to  the  armature  and 
therefore  remain  stationary  in  space  the  same  as  the  field-poles. 

Equations  of  Synchronous  Motors.  Analytical  Theory.  We  have 
just  examined  the  phenomena  of  synchronous  motors  from  a  physical 
point  of  view.  We  shall  now  represent  them  analytically,  according 
to  the  theory  first  expounded  by  Dr.  J.  Hopkinson,  but  with  a  few 
modifications  in  form.  We  shall  suppose  with  him  that  the  E.M.F.'s 
and  currents  follow  the  sinusoidal  law,  and  that  the  reactances  of  the 
machine  are  constant. 

Let  IBS  suppose,  then,  a  single-phase  A.C.  generator  and  motor, 
defined  by  their  induced  E.M.F.'s,  their  resistances,  and  their  mean 
inductances,  which  are  all  supposed  constant. 

Let  T"=the  duration  of  the  period; 

(JL)  =  —  =  the  speed  of  pulsation  of  the  currents; 

ei  and  e%  =  the   instantaneous  values  of    the  generator    and  motor 

E.M.F.'s  respectively,  at  the  instant  /; 

EI  and  £2  =  the  effective  values  equal  to  the  amplitudes  of  the  sine- 
functions,  i.e.,  the  maximum  value,  divided  by  \/2; 

-^  =  the  phase-difference  between  e\  and  e2; 
#  =  the    angle    of    lag    (phase-difference)    corresponding    to 


16  SYNCHRONOUS  MOTORS 

R  and  Z,=the  resistance  and  inductance,  respectively,  of  the  total 

circuit,  of  the  two  machines; 
i  =  the  instantaneous  value  of  the  current; 
7  =  the  effective  value  of    the  current,  equal  to    the  maxi- 
mum value  divided  by  \/2. 

Let  us  suppose  the  conditions  of  stability  to  be  unknown  and  let 
us  seek  to  ascertain  how  two  alternators  connected  in  series  will  operate. 

The  two  sine-functions  of  the  E.M.F.  represented  by  the  curves 
e\  and  e^  in  Fig.  n,  may  be  formulated  by  the  equations, 


sn 


62=          2  sn 


in  (wt-\  —  J; 
in  I  cot  --  J  ; 


in  which  d  designates  the  angular  distance  between  the  actual  position 
of  e\  and  the  position  of  opposition  of  £2. 

The  E.M.F.  which  is  acting  in  the  circuit  is  equal  to  the  algebraical 
sum  of  the  opposing  E.M.F.'s. 


sin  (  cot  +  —  )  —  E<>  \/2  smlcot  --  ). 
V        2/  V        27 


From    this  the  current,  i,  may  be  deduced,  by  the  well-known 
differential  equation, 


mlcut  --  ).        (A) 

\  2/ 


sm 

dt 

In  this  equation  let  i=X  sin  wt+Y  cos  cut. 

If  this  value  be  substituted  in  the  equation,  the  values  of  X  and  Y  can 
be  determined  by  making  the  coefficients  of  the  sine-terms  and  of  the 
cosine-terms  successively  equal  to  zero.  We  can  then  obtain,  by 
differentiation,  substitution,  etc.,1  the  following  value  for  i: 


R2+OJ2L2 


f.    /        B\     coL        I        6\-\    '      (^ 
^sm^__j__cos^__jj 

1  See  Appendix  A. 


GENERAL  PRINCIPLES  OF  SYNCHRONOUS  MOTORS     17 
This  may  also  be  written, 


E1     _    Tf 

wherein  tan  8  =  -=?  —  ^  tan  d; 


coL 
tan  r=-. 

In  this  equation  /?=the  phase-angle  of  the  resultant  E.M.F.  and  ;-=the 

supplemental  phase  -difference  of  the  current  measured  from  this  E.M.F. 

In  the  simple  particular  case  where  EI  =E2  this  expression  reduces  to 

,    0 
sin  — 

i  =  2E\/2    2       2-^  [R  cos  a>t+a)L  sin  wt]; 


or,  since 


we  will  have 


R  coL 

=  >s  r'  *=sm  r' 


/\ 

sin  —  cos  (ojt— 


i.e.,  the  current  will  have  the  effective  value 


and  will  be  out  of  phase  by  the  angle  7-  with  respect  to  the  resultant 
E.M.F.,  which  is  itself  out  of  phase  —  with  respect  to  the  mean  of  e\ 

and  62.  This  result  is  easily  interpreted  in  Fig.  n,  by  drawing  the 
resultant  curve  ei+e2)  obtained  by  taking  the  difference  of  the  ordinates 
of  the  first  two  curves.  It  will  be  seen  that  the  curve  has  actually  a 

phase-difference  equal  to  —  with  regard  to  the  mean  of  e\  and  e2,  and 


18 


SYNCHRONOUS  MOTORS 


that  it  increases  with  the  phase-difference  of  e%  with  respect  to  e\.     In 
consequence  of  the  lag,  f,  of  the  current,  measured  with  respect  to  this 

re.sultant  E.M.F.  (when  f  is  near  —  in  value  J,  it  will  be  seen  that  the 

current  is  approximately  in  phase  with  this  mean  value;   it  would  be 
completely  so  if  there  were  no  resistance-losses. 


FIG.  ii. 

The  power-outputs  of  the  two  machines  will  be  obtained  by  multi- 
plying the  instantaneous  current  i  by  the  E.M.F.'s.  e\  and  e^.  For 
example,  in  the  case  where  E1=E2,  we  have 


4E2 


0\ 


.  e  r  .   /    0\        /          0\i 

sm  —   sm  (  r  +  —    +sm  [20*1—  r-\  —  II. 

*  2[          V        2/  \  2/J 

Likewise,  on  multiplying  by  —  e2,  we  will  have 

+  2£2        ,    0[  .  -/      0\  t        (  0\] 

p2  =    ,  —        —  sin  —   sin  [r  --    +sm  I  2ojt—r  --  )    . 
L2        2[       \'      2/  \  2/J 


These  equations  show  that  the  power  is  not  constant,  in  either  case, 
but  pulsating,  i.e.,  it  presents  variations  of  frequency  =  2  T,  as  repre- 
sented in  Fig.  1  1  . 


GENERAL  PRINCIPLES  OF   SYNCHRONOUS  MOTORS     19 

These  variations  constitute  sine-functions  having  pulsations  twice 
as  rapid  as  those  of  the  current,  which  have  for  their  axes  the  horizon- 
tal lines  (Pi,  P2,)  corresponding  to  the  mean  powers  given  by  the  first 
terms  within  the  brackets  in  the  following  equations: 


sin  —  sm 

2 


K) 


E2 


The  very  small  difference  between  PI  and  P2  represents  the  loss 
by  resistance  (Joule  effect).  The  axis  of  the  curve  PI  is  therefore 
a  little  more  above  the  axis  of  zero  power  than  the  axis  of  symmetry 
of  the  curve  P2  is  below  it. 

The  torque  could  be  obtained,  in  each  case,  by  dividing  the  power 
by  the  angular  velocity.  These  expressions  show  that  the  current 
increases  with  6  until  0  equals  TT,  but  the  torques,  which  equal  zero  so 
long  as  the  lag  6=  zero,  increase  with  6  only  until  the  value  O  =  Y;  and 
they  then  decrease. 

Stability  will,  therefore,  exist  only  with  6  <  Y  having  for  its  axis  the 
exact  opposition  of  E.M.F.'s. 

The  solution,  in  the  case  where  EI  is  different  from  E2,  will  be 
obtained  in  an  analogous  manner,  by  forming  the  products  e^i—  e2i; 
and  it  would  still  give  pulsating  values  for  pi  and  p2;  but,  since  these 
caculations  are  uselessly  complicated,  we  will  pass  them  by  and  turn 
to  more  simple  methods. 

Case  of  Symmetrical  Polyphase  Motors.  In  the  case  of  polyphase 
motors  the  same  considerations  and  equations  remain  applicable  to  each 
of  the  symmetrical  circuits,  if  only  care  be  taken  to  include,  in  the 
self-induction  of  these  circuits,  their  mutual  induction  effects.  By 
reason  of  symmetry  itself,  the  result  is  obtained  by  a  simple  in- 
crease of  the  coefficient,  L,  in  a  ratio  which  depends  on  the  number 
of  phases  in  the  machine. 

The  currents  are,  in  general,  approximately  equal  in  effective  values 
in  the  different  circuits,  if  there  are  no  defects  in  construction.  It 


20  SYNCHRONOUS  MOTORS 

is  sufficient,  therefore,  to  apply  the  reasoning  to  one  of  the  circuits 
only,  and  to  multiply  the  power  by  the  number  of  circuits. 

It  should  be  noted,  moreover,  that,  in  these  motors,  the  pulsations 
of  the  power  derived  from  the  different  circuits,  occurring  at  different 
intervals,  compensate  each  other,  from  the  standpoint  of  the  total 
power,  which  becomes  constant.  This  is  easily  shown  by  taking  the 
sum  of  the  powers.  For  example,  in  a  three-phase  motor,  the  pulsa- 
at  the  instant  2  cot  will  have  the  form 

A  sin 


A      '       (  0         2K\ 

A  sin  (2cot~-f J; 

/  0     47r\ 

A  sin     20)t—  r . 

V          '      2       3/ 


It  is  known  that  the  sum  of  the  sines  of  three  angles  differing  from 
each  other  by  120°  is  identically  equal  to  zero.  Therefore  the"  result- 
ant" pulsation  is  equal  to  zero.  The  same  thing  would  be  true  for 
any  number  whatever  of  equidistant  and  symmetrical  phases. 

The  result  is  that  the  torque  is  constant  (to  a  sufficient  degree  of 
approximation  for  this  theory,  which  neglects  the  higher  harmonics 
of  the  field-distortions),  whereas,  in  a  single-phase  motor,  it  undergoes 
heavy  periodical  variations.  In  the  latter  case,  the  inertia  of  the  armature 
plays  the  role  of  a  flywheel  storing  and  restoring  energy  twice  during 
each  period;  but  if  the  inertia  is  insufficient,  the  velocity  of  the  armature 
will  experience  slight  variations  which  will  greatly  interfere  with  the 
stability  of  operation.  Polyphase  motors  are,  in  this  respect,  superior 
to  single-phase  motors;  they  are  also  superior  to  them  in  being  lighter 
for  a  given  output  (less  weight  per  kilo-volt-ampere)  and  also  in 
having  higher  efficiency. 

Graphical  Representation  of  Operative  Conditions.  Blakesley's 
Method.  Mr.  Blakesley  was  the  first  to  apply  Fresnel's  method  of 
vectors  to  the  study  of  alternating  currents.1 

The  following  principles  constitute  the  basis  of  Fresnel's  method: 

i.  Any  sinusoidal  function  can  be  represented  in  magnitude  and 
in  phase  by  a  vector,  or  a  segment  of  a  right  line,  whose  length  is  pro- 
portional to  the  amplitude  of  the  function  and  whose  phase  is  meas- 
ured by  an  angle  reckoned  from  some  other  vector  serving  as  a  point 
of  origin. 

1  Blakesley,  Alternating  Currents  of  Electricity,  1885 


GENERAL  PRINCIPLES  OF  SYNCHRONOUS  MOTORS     21 


2.  The  addition  or  subtraction  of  sinusoidal  functions  may  be  made 
on  the  graph  by  a  geometrical  addition  or  subtraction  of  the  vectors 
of  these  functions. 

3.  The  mean  product  of  two  functions  is  equal  to  the  work  done 
by  one  of  the  vectors  on  the  other,  i.e.  it  is  equal  to  the  area  of  the 
triangle  constructed  on  vectors  which  are  equal  to  the  effective  values 
of  the  variables,  or  to  half  this  area  when  the  triangle  is  constructed 
with  the  amplitudes  of  the  sinusoidal  functions. 

This  method,  with  which  the  reader  is  supposed  to  be  familiar, 
has  been  applied  by  Blakesley  to  the  problem  of  synchronous  motors 
in  the  following  manner: 

Let  us  represent  the  amplitudes  of  the  E.M.F.'s.  of  the  generator 
and  of  the  motor,  E1  \/2,  and  E2  V^,  re- 
spectively, by  means  of  two  vectors, 
OA  and  OB,  (Fig.  12),  having  between 
them  the  angle  71+6.  These  E.M.F.'s 
are  approximately  opposed  to  each  other, 
as  we  have  seen  in  what  precedes. 

Let  OB'  be  equal  but  opposed  to  OB. 
The  E.M.F.  eN/2,  which  is  the  resultant 
of  £1^2  and  £2^2,  will  be  represented 
by  the  vector  B'A  which  is  equal  to  the 
resultant  of  OA  and  OB. 

Let  us  again  designate  by  f  the 
phase-angle  between  the  current  and 
the  resultant  E.M.F.  denned  by  the  re- 
lation 

ajL 
*«  r -IP 

and  let  this  angle  be  drawn  with  respect  to  the  point  A.  The  direction 
CA  will  represent  the  current  which  is  in  phase.  Let  CA  be  the  pro- 
jection of  B'A .  We  will  have 

CA=WA  cos =eV^cos 


or 


CA 

_ . 


The  length  of  CA  is  therefore  proportional  to  7. 


22  SYNCHRONOUS  MOTORS 

The  power  supplied  by  the  generator  to  the  motor  will  be 
Pl=ElI  cosOAD 

JDAXCA 

2R 
In  like  manner,  the  electric  power  consumed  by  the  motor  will  be 

JDCXCA 

The  power  lost  by  ohmic  resistance  (Joule  effect)  is  equal  to  the 
difference,  or  to 

CAXCA 


The  diagram  therefore  determines,  by  the  measurement  of  simple 
lines  or  areas,  the  current  strength  and  the  power  of  each  of  the 
two  machines,  for  each  value  of  the  phase-angle. 

Let  us  suppose  that  the  vector  OA  (Fig.  13)  is  stationary,  and  that 


the  phase-angle,  0,  varies.  The  point  Bf  will  describe  a  circle  of  radius 
OB'-,  the  point  D  describes  a  circle  constructed  on  OA  as  a  diameter; 
finally,  it  is  easily  seen  that  the  point  C  describes  a  third  circle  having 
for  its  center  and  for  its  radius  the  projections  of  O  and  of  OG  on  a 
right  line,  AF,  making  the  angle  f  with  OA.  These  three  circles  hav- 
ing been  drawn,  the  conditions  of  operation  can  be  followed  on  the 
diagram.  Mr.  Blakesley  has  thus  determined  the  limits  of  stability 
and  the  maximum  electrical  efficiency. 


GENERAL  PRINCIPLES  OF  SYNCHRONOUS  MOTORS     23 

But  even  for  easy  questions  the  use  of  this  diagram  is  too  com- 
plicated; the  same  thing  is  true  of  the  diagrams  devised  by  Kapp,1  after 
Blakesley,  and  based  on  the  same  principles.  The  author  has  also 
given2  a  more  convenient  method  of  representation  of  power  by 
circles  which  cut  off  segments  directly  from  the  lines  OA  and  OB. 

In  all  that  follows,  we  will  give  preference  to  another  graphical 
method,  still  more  simple,  which  will  be  explained  in  Chapter  II. 

Equation  of  the  Synchronous  Motor  by  the  Method  of  Complex 
Variables.  Finally,  to  complete  the  review  of  the  different  methods 
of  study  proposed  or  used  for  the  representation  and  study  of  the 
phenomena,  attention  should  be  given  to  the  equations  which  translate 
the  preceding  diagram  by  the  method  of  complex  variables,  already 
applied  to  alternating  current  problems,  in  the  exponential  form,  by 
Oberbeck,  Cornu,  Chaperon,  etc.,  and  in  the  linear  form  by  Kennelly 
Steinmetz,  Guilbert,  etc.  It  is  merely  an  application  of  the  ordinary 
geometrical  representation  of  imaginary  quantities. 

We  will  employ  the  notation  of  Steinmetz  modified  by  M.  Guil- 
bert (La  Lumicre  Electrique,  Vol.  L,  p.  451,  and  VEdairage  Electrique, 
Vol.  XIV,  p.  69),  i.e.,  by  expressing  the  impedance  as  (r+si)  instead 
of  (r  —  si),  the  latter  expression  being  scarcely  logical,  since  the  reactance 
acts  in  the  same  direction  as  a  resistance  to  reduce  the  current.  It 
thus  becomes  possible  to  retain  the  ordinary  axes.3 

(i)  Let  OX,  OY  (Fig.  14)  represent  two  rectangular  co-ordinate 
axes.  Let  OA  be  a  vector  repre- 
senting the  sinusoidal  function  whose 
projections  are  x  and  y.  Geomet- 
rically, the  vector  A  is  defined  by  the 
two  projections,  and  analytically  it 
is  represented  by  a  single  imaginary 
value 

x+yj  =  (A  cos  6}  +  (-4  sin  0)j, 

in    which  ;   is  the    imaginary    sym- 
bol V^T. 

All  the  harmonic  functions  may  be  also  represented  by  imaginary 
quantities. 

1  Electrical  Transmission  of  Energy.     London,  Whittaker,  1892. 

2  La  Lumiere  Electrique,  Vol.  XLV,  1892,  p.  415,  and  Bulletin  de  la  Societe  des 
Electriciens,  1893. 

3  Compare  with  C.  P.  Steinmetz,  Theory  of  the  Synchronous  Motor  (Transactions 
of  the  American  Institute  of  Electrical  Engineers,  Oct.  17,  1894) 


24  SYNCHRONOUS  MOTORS 

(2)  These  quantities  are  added  or  subtracted  the  same  as  real 
quantities;    the   resultant   quantities  indicate   immediately,   by   their 
real  and   imaginary    portions,   the  magnitude   and   the  phase  of  the 
resultant  vector. 

(3)  The  multiplication  of  a  complex  quantity  by  a  real  quantity 
only  changes  the  dimension  of  the  vector,  without  changing  its  phase. 
On  the  contrary,  the  multiplication  by  an  imaginary  portion,  such  as 
jbj  modifies  not  only  the  magnitude  of  the  vector,  which  thereby  becomes 
multiplied  by  b,  but  it  also  modifies  its  phase,  which  is  then  made  to 

advance  by  the  amount  —  .     In  fact,  we  have 


i.e.,  the  vector  OA  is  then  replaced  by  the  vector  OA'. 

(4)  From  this,  it  follows  that  the  E.M.F.  absorbed  by  an  imped- 
ance, z,  composed  of  a  resistance,  r,  and  of  a  reactance,  s,  in  series, 
through  which  passes  a  current 


is  obtained,  in  magnitude  and  phase,  by  forming  the  product 


(5)  The  work  done  by  an  E.M.F.  OB  having  the  imaginary  value 

a+jb, 

and  a  current  OA  having  the  imaginary  value 

x+jy, 

is  easily  obtained  by  decomposing  it  into  the  work  of  each  of  the  com- 
ponents a  and  b  of  this  E.M.F.  The  first  component,  a,  does  no  work 
on  the  component  y  of  the  current  with  which  it  is  "  in  quadrature" 
but  it  is  "in  phase  "  with  the  component  x;  likewise  b  is  in  quadrature 
with  x  but  in  phase  with  y.  The  power  required  is  therefore  the  sum  of 
the  products  of  the  analogous  coefficients  of  E  and  of  7.1  We  will  have 

ax-}-  by. 

1  Use  can  also  be  made  of  other  equivalent  rules.  A  rule  is  given  by  M.  Janet 
(Eclairage  Electrique,  18.  Dec.,  1897,  p.  530)  under  the  following  form:  To  find 
the  power,  j  is  changed  into  —  ;  in  either  the  expression  for  the  current  strength 


GENERAL  PRINCIPLES   OF  SYNCHRONOUS  MOTORS     25 

Let  us  apply  this  notation  to  the  generator  and  motor  already  con- 
sidered. Let  us  direct  the  E.M.F.  E1  of  the  generator  along  the  axis 
OX.  E2,  lagging  in  phase  by  the  angle  6,  measured  from  the  opposite 
direction,  will  have  the  form 

E2  (cos  0-j  sin  6}. 
The  imaginary  impedance  is 

R+ajLj. 
The  resultant  E.M.F.  being 

El-  (E2  cosd-jE2  sin  0), 
the  current  will  be  obtained  by  taking  the  quotient 

T    Ei-E2Cosd+jE2smO  '/J.-JT     •    m   R-a>Lj 

K+uLi  =(£i-£2  cos  0  +y£2  sin  0)  - 

Separating  the  real  and  imaginary  portions,  we  have 


if     R(Ei  —  E2  cos  6)  +ajLE2  sin  0 
~R*+u2L?  {  +  j[RE2  sin  d-ajL(E1-E2  cos  0)] 

This  equation,  when  transformed  into  finite  values,  and  taking  z  to 
represent  the  impedance,  gives 

Z2/2  =  [R(Ei  -  E2  cos  6)  +ajLE2  sin  6]2  +  [RE2  sin  0-ajL(El  -  E2  cos  0)]2 

This  equation  defines  the  relation  between  I,  E\t  E2,  and  6  in  a  syn- 
chronous motor. 

The  mean  power,  PI,  is  obtained  by  simply  multiplying  the  real 
portion  by  EI    thus: 

cos  0-"L  sin 


cos  7—^2  cos 


or  the  expression  for  the  E.M.F.  The  product  is  calculated,  and  then  only  the 
real  portion  thereof  is  taken.  M.  Guilbert  has  also  given  the  following  rule  (E.  E., 
10  Mar.,  1900,  p.  361):  Change  the  sign  of  the  variable  which  is  out  of  phase 
and  the  real  portion  gives  the  power,  with  its  sign. 


26  SYNCHRONOUS  MOTORS 

As  for  P2,  it  can,  according  to  the  rule  already  stated,  be  expressed  as 
follows: 

E2  cos  6[R(Ei-E2  cos  6)  +a)LE2  sin  6]  } 


R2+u2L2  \  -E2  sin  6[RE2  sin  0-ajL(E1-E2  cos  &)] 
E'2     _  [Ei  cos  (r-  6)  -  E2  cos  r]. 

[ 


This  expression  could  have  been  written  directly,  by  symmetry. 

Such  is  the  method  of  imaginaries  for  establishing  the  fundamental 
equations.  It  will  be  seen  that  the  calculations  thereby  made  are  more 
simple  than  with  the  analytical  method  of  Hopkinson,  because  they 
give,  immediately,  the  general  solution,  and  do  not  require  any  inte- 
gration. In  reality,  however,  complex  variables  only  constitute  an 
artifice  for  writing  down  the  results  of  the  graphical  method.  In  place 
of  detailed  reasonings  which  are  rendered  more  precise  by  means  of 
diagrams,  they  substitute  algebraical  operations,  which  are  effected 
mechanically,  without  benefit  in  helping  the  mind  to  understand  the 
physical  phenomena.  In  all  that  follows  we  will  therefore  adopt  the 
graphical  method. 

Excitation  of  Synchronous  Motors.  Synchronous  motors  may  be 
excited  in  three  ways: 

(1)  As  in  the  case  of  alternators,  it  is  possible  to  use  the  current 
produced    by  a  small  exciter-dynamo    mounted  on  the  same  shaft. 
The  machine  is  put  in  operation  by  one  of  the  methods  indicated  later, 
or  else  by  running  the  exciter-dynamo  as  a  motor,  by  means  of  current 
from  a  storage  battery.     This  method  of  excitation  is  advantageously 
used  for  large  motors. 

(2)  To  avoid  the  complication  of  the  exciter-dynamo,  especially  for 
small  motors,  the  machine  is  often  made  self-exciting  by  means  of  corn- 
mutated  currents,  by  sending  through  the  field-circuit  a  portion  of  the 
current  obtained  from  the  mains.     The  oldest  method  of  commutating 
this  current  consisted  in  using  a  simple  shell-commutator  by  which  the 
current  was  reversed  at  each  change  of  polarity.     Such  was  the  Ganz 
commutator  shown  diagrammatically  in  Fig.  15  for  a  six-pole  motor. 
The  inducing  circuit  connects  with  three  segments  />,  p',  p"  connected 
in  parallel  with  one  pole,  and  the  other  end  of  the  circuit  connects 
with  three  other  segments  n,  n',  n",  which  are  connected  in  parallel 
with  the  other  pole.     The  alternating  current  to  be  commutated  is 


GENERAL  PRINCIPLES  OF  SYNCHRONOUS  MOTORS     27 


brought  by  two  double  brushes  bb',  bib',  which  can  be  shifted  to  a 
suitable  extent  by  means  of  the  collar  supporting  them,  the  object 
being  to  reduce  sparking  to  a  minimum.  The  use  of  double  brushes 


FIG.  15. 


FIG.  16. 


has  for  its  object  to  reduce  the  sparking  by  short-circuiting  the  field 
windings  at  the  time  when  the  induced  current  is  reversing;  the  extra 
current  which  is  then  produced  in  the  exciting  coils  prolongs  the  cur- 
rent in  the  field-circuits  and  diminishes 
the  undulations  of  the  inducing  mag- 
netic flux.  Nevertheless,  these  fluctu- 
ations are  quite  perceptible  and 
interfere  very  much  with  the  efficiency 
and  the  stability  of  the  machine. 
These  commutating  arrangements 
can  only  be  used  for  low  voltages. 
For  this  reason,  when  the  voltage  at 
the  terminals  of  the  motor  exceeds  no 
volts,  the  commutator  is  supplied  by  the 
secondary  winding  of  a  transformer 
which  is  connected  in  multiple  with  the 

mains  or  else  by  a  special  low-voltage  circuit  wound  on  the  same  arma- 
ture, as  in  the  case  of  the  old  Westinghouse  alternators  (Fig.  17). 

Commutated  currents  can,  theoretically,  be  used  for  excitation 
either  in  series  or  in  shunt  (even  compound)  as  indicated  in  the  dia- 
gram of  Figs.  1 8  and  19.  The  advantages  and  disadvantages  of 
these  various  systems  are  the  following:  1 

Series-excitation  allows  the  use  of  large  wires  for  the  field-coils, 
and  insures  coicidence  in  phase  between  the  currents  of  the  inducing 

1  A  more  detailed  study  of  these  "  commutated "  excitations  will  be  found  in 
an  article  on  synchronous  motors  by  the  author,  in  La  Lumiere  Electrique,  1892, 
Vol.  Ill,  pp.  465,  466,  etc. 


FIG.  17. 


28 


SYNCHRONOUS  MOTORS 


and  induced  circuits  during  the  operation  of  starting,  which  operation 
is  thereby  facilitated.  It  has,  however,  the  great  objection  of  making 
the  induced  E.M.F.  vary  with  the  load,  whereas,  according  to  theory, 
the  most  advantageous  E.M.F.  is  fixed  a  priori,  and  should  remain 
practically  independent  of  the  load.  With  series-excitation,  strong 
wattless  currents  occur  at  very  light  and  very  heavy  loads  (which  may, 
it  is  true,  be  reduced  by  shunting  the  field-coils,  as  shown  in  Fig.  19), 
and,  besides,  there  is  a  lack  of  stability  of  operation  at  light  loads 


FIG.  19. 


because  the  field  is  then  too  weak.     Morevoer  series-excitation  causes 
very  bad  sparking  at  the  brushes. 

Shunt-excitation  has  the  advantage  of  being  constant  and  easily 
regulated;  but  it  has  another  objection,  that  of  necessitating  change 
of  position  of  the  diameter  of  commutation  with  respect  to  the  phase 
of  the  current  to  be  commutated.  With  normal  load,  the  position  of 
the  brushes  can  be  easily  regulated  in  such  a  way  that  the  commutation 
may  occur  exactly  when  the  alternating  current  to  be  commutated  is 
passing  through  the  zero-point;  but  if  the  load  increases,  the  lagging 
of  the  armature  behind  the  field  increases,  without  being  followed  by 
the  phase  of  the  current  to  be  commutated.  The  same  thing  occurs 
during  any  accidental  oscillation  of  the  speed  of  the  armature  (rotor). 


GENERAL  PRINCIPLES  OF  SYNCHRONOUS  MOTORS     29 

Now,  as  Fig.  20  shows,  the  mean  strength  of  the  commutated  current 
is  proportional  to  the  algebraical  sum  of  two  areas,  one  of  which  (Si) 
is  positive,  and  the  other  (52)  is  negative.  The  former  diminishes  and 
the  latter  increases,  when  the  lag  increases,  the  effect  being  a  rapid 
decrease  in  the  excitation.  If  we  let  /o  =time  at  which  the  commuta- 

tion occurs;  if  T=the  period,  and  271-^=  the  lag  in  question,  the 
exciting  ampere-turns  will  be  given  by  the  following  equation: 

A^av.  =  ~=r  I          i0  sin  (  2n—  —  B)dt  =  -Ni0 
1  Jt0  \    T       /       n 


cos  271 


i.e.,  they  are  reduced  in  the  proportion  indicated  by  the  factor  cos  —  ~ 
as  compared  with  their  value  when  the  commutation  occurs  at  the 


FIG  20. 

proper  moment  of  passing  through  zero  (at  which   time  the  factor 
cos-—-  equals  unity).     It  is  necessary,  therefore,  to  vary  the  position 

of  the  brushes  with  the  load.  This  cannot  be  done,  however,  during 
the  fluctuations  of  load,  and  it  is  conceivable  that  this  decrease  of 
excitation  under  the  influence  of  the  lag  may  then  greatly  reduce  the 
stability  of  operation.  It  appears,  therefore,  that  the  method  of  exci- 
.  tation  by  commutated  currents  is  a  process  open  to  criticism. 

Advantage  might  be  taken  of  the  use  of  the  transformer  in  Fig. 
21,  as  the  author  showed  in  1892  (Lumiere  Electrique,  loc.  cit.),  to 
compound  the  excitation  in  such  a  way  as  to  cause  the  inducing 
flux  to  increase  with  the  load,  and  thus  produce  an  increase  in 
stability.  It  would  be  sufficient,  to  this  end,  to  wind  on  the  same 
transformer  a  second  primary  coil  with  a  large  conductor  through 
which  the  armature  or  rotor  current  passes. 

(3)  Another  method  of  commutating  the  current,  which  is  pref- 
erable to  the  preceding  one,  consists  in  combining  with  the  armature- 


30 


SYNCHRONOUS  MOTORS 


winding  a  continuous-current  commutator  from  which  the  exciting 
current  is  collected  by  means  of  two  brushes.  Such  is  the  system  of 
the  Fort  Wayne  Company  in  America,  and  of  the  Socie*te  1'Eclairage 
Electrique  (Labour  system)  in  France.  The 
commutator  need  be  only  of  small  dimensions, 
proportional  to  the  current  which  it  delivers, 
unless  it  is  also  used  in  starting  the  motor. 
When  the  voltage  of  the  armature  is  too 
high  or  when  it  is  desirable  to  simplify  the 
connections  of  the  main  circuit,  it  is  suffi- 
cient to  wind,  on  the  armature,  a  small  exciter 
winding  connecting  with  a  small  commutator, 
which  serves  solely  for  the  purpose  of  excitation. 
This  method  of  excitation,  which  is  the 
simplest  of  all,  unfortunately  still  presents 
several  objections.  It  is  applicable  only 
to  motors  having  continuous  current  wind- 
ings and  a  rotating  armature;  it  cannot  be 

adopted  in  the  case  of  high  voltage  machines  or  alternators  having 
a  stationary  induced  winding;  it  does  not  lend  itself  to  automatic 
regulation  of  the  inducing  field;  finally,  (and  this  is  the  most  serious 
objection),  when  there  are  wattless  demagnetizing  currents  in  the  motor, 
from  any  cause,  the  E.M.F.  at  the  brushes  and  the  excitation  are  both 
weakened  in  consequence.  Now,  that  is  precisely  what  occurs  when 
the  motor  experiences  a  reduction  in  speed  in  consequence  of  an  over- 
load. The  stability  of  operation,  which  decreases  with  the  excitation, 
will,  therefore,  be  diminished  when  the  load  increases.  The  oscilla- 
tions due  to  •  improper  operation  are  therefore  exaggerated  by  this 
reaction. 

The  self-excitation  of  synchronous  motors  must,  therefore,  be 
studied  with  care,  and  its  application  is  limited  to  small  motors.  For 
larger  machines  the  use  of  a  separate  exciter  is  the  most  satisfactory 
plan. 


CHAPTER  II 
DETAILED   STUDY   OF   OPERATION   WITH   NORMAL   LOAD 

I.   PRINCIPLES    OF   THE   ELEMENTARY   THEORY 

Notation.     Let 

R  =  the  resistance  of  the  total  circuit  of  the  motor  (including  the  motor, 
the  line,  and  the  armature  of  the  generator,  in  the  case  of  an 
electric  transmission  system.  (The  armature  resistance  of  the 
alternators  is  supposed  to  be  increased  from  50  to  100  per  cent 
in  such  a  way  as  to  take  into  consideration  the  eddy  currents 
produced  by  the  armature  in  the  pole-pieces); 

L=the  mean  inductance  of  the  motor; 
/  —  that  of  the  external  circuit  (including  line  and  generator); 

T"=the  period;  oj  =  —  =the  velocity  of  pulsation; 

X=the  reactance  of  the  total  circuit,  including  the  motor;  X=w(L-\-l)-, 
Z=the  impedance  of  the  total  circuit,  including  the  motor: 


£1=- the  effective  external  E.M.F.,  i.e.,  the  E.M.F.  obtained  from  the 
line  (or  induced  in  the  generator,  in  the  case  of  an  ordinary 
simple  transmission  system).  This  E.M.F.  is  supposed  constant, 
unless  otherwise  stated,  in  all  that  follows; 

E2=the  effective  internal  E.M.F.  induced  in  the  synchronous  motor, 
such  as  it  would  be  measured  on  open  circuit  with  the  same 
excitation.     (E2  should  not  be  confounded  with  the  resultant 
induced  or  effective  E.M.F.  produced  by  the  resultant  field  of 
the    field    ampere-turns  and  of  the  armature-reaction.      The 
latter  will  be  designated,  later,  by  the  symbol  £2-) 
e=the  resultant  E.M.F.  of  the  two  E.M.F's.  EI  and  E2; 
7  =  the  current  in  the  armature  of  the  motor; 
PI  =the  electric  power  produced  by  the  E.M.F.  EI  ; 

31 


32  SYNCHRONOUS  MOTORS 

P%  =  the  electric  power  generated  in  the  armature  by  the  current,  and 

counted    positively   in   the   same  direction   as  the  mechanical 

power  obtained  from  the  motor; 

#=the  phase-angle  between  the  E.M.F's.  E\  and  E2', 
(j)=the  phase-angle  between  the  current,  I,  and  the  E.M.F.  E2,  of 

the  motor; 

</>=the  phase-angle  between  the  current,  7,  and  the  external  E.M.F.,  E\. 
f  =  an  auxiliary  angle  representing  the  phase  -angle  existing  between 

the  current  and  the  resultant  E.M.F.,  e,  which  produces  it  in  the 

circuit.  '  It  is  defined  by  the  known  relation, 

.          X 
sm  7-=^-, 
Z 

R 


(i)  tanr  =  —  ^=  —  -=X\       whence 


=  an  analogous  angle,  obtained  by  neglecting  the  reactance  of  the 
motor.     It  is  defined  by  the  relation, 


The  standard  diagram  which  has  already  been  referred  to  is  not 
easy  to  apply,  partly  because  a  complicated  construction  is  necessary 
for  each  load  to  give  the  current-strength  in  magnitude  and  in  phase, 
and  partly  because  the  vectors  of  the  current-strength  become  too 
small,  inasmuch  as  the  E.M.F.  may  attain  thousands  of  volts  while  the 
currents  are  of  a  few  amperes  only.  The  geometrical  constructions 
deduced  from  this  diagram  by  other  authors  are  extremely  complicated. 

The  author  devised,  some  years  ago,1  for  the  study  of  problems 
of  this  kind,  a  so-called  "  bipolar  "  diagram,  which  avoids  these  two 
objections  and  at  the  same  time  gives  the  theory  an  almost  childlike 
simplicity,  and  renders  graphical  calculation  sufficiently  accurate. 

Principle  of  Bipolar  Diagrams.  The  principle  of  this  method  con- 
sists in  taking  different  axes  of  reference  and  different  scales  for  the 
currents  and  for  the  E.M.F's.,  instead  of  taking  the  same  axis  of 
reference  and  the  same  scale  for  both,  as  is  usually  done.  The  axes 
and  scales  may  then  be  selected  for  the  current  in  such  manner  that  the 

1  Theorie  des  moteurs  synchronies.  Lahure,  publ.  1895.  L1  Industrie  Electrique, 
Feb.,  1895- 


DETAILED   STUDY  OF  OPERATION  WITH  NORMAL  LOAD  33 

current  can  be  represented  in  magnitude  and  in  phase  by  the  same 
vector  as  the  resultant  E.M.F.  of  the  circuit. 
As  regards  the  magnitude,  since  we  have 

e=ZI, 

it  suffices  to  take,  for  the  current  7,  a  scale  of  amperes  Z  times  greater 
than  the  scale  of  volts  used  for  the  E.M.F.  values. 

As  to  the  axis  of  reference,  its  choice  depends  on  the  axis  selected 
for  the  E.M.F.  values. 

The  bipolar  diagram  may  therefore  be  established  in  two  distinctly 
different  manners,  according  to  whether  it  be  the  vector  of  the 
generator  E.M.F.,  Ej  or  that  of  the  motor,  E2,  which  is  taken  as 
the  fixed  line. 

We  will  indicate,  successively,  these  two  types  of  diagrams  which, 
in  many  cases,  do  not  serve  exactly  the  same  purpose,  as  will  be  seen 
later. 

Bipolar  Diagram  of  the  First  Kind.  Motor-Vector  £2  taken  as 
Fixed  Axis.  Let  us  suppose  the  effective  values  EI  and  E2  of  the 
E.M.F's.  taken  with  the  signs  which  they  have  when  one  follows  the 
circuit  formed  by  the  motor,  the  line,  and  the  generator.  It  is  known, 
by  experience,  that  the  E.M.F's.  are  very  nearly  in  opposition.  The 
two  corresponding  vectors  can,  therefore,  be  represented  by  two  right 
lines  OA2  and  OAi,  drawn  in  different  directions  (as  indicated  by 
the  arrow-heads)  and  making,  a  very  small  angle,  6,  with  each  other 
(Fig.  22).  (This  angle  indicates  the  phase-difference  between  the  two 
E.M.F's.,  as  measured  at  the  terminals  of  the  two  machines  when 
coupled  in  parallel.) 

When  under  load,  the  generator  tends  to  lead  with  respect  to  the 
motor.  The  vector  EI  will,  therefore,  turn  in  the  positive  direction 
which  is  conventional  in  mechanics,  i.e.,  in  the  direction  contrary  to 
the  hands  of  a  watch. 

The  resultant  E.M.F.  e,  acting  in  the  circuit,  is  represented  by  the 
vector  AiA2,  which  is  the  resultant  of  EI  and  E2.  By  projecting 
AiA2  on  a  right  line,  A2F,  which  makes  the  angle  7-  backward  from 
A iA2,  we  obtain  the  vector  A2F=RI,  representing  the  component  that 
is  in  phase  with  the  E.M.F.  EI.  The  angle  DA2A\,  therefore  measures 
the  phase-difference  of  the  current  with  respect  to  the  E.M.F.  E2  with, 
reversed  sign,  i.e.,  measured  at  the  terminals,  the  same  as  the  E.M.F. 
of  the  generator.  In  order  to  be  able  to  take  A iA2  as  equal  in 


34  SYNCHRONOUS  MOTORS 

magnitude  and  in  phase  to  the  vector  of  the  current,  it  suffices,  there- 
fore, to  attribute  to  it,  as  an  axis  of  reference  for  phases,  a  right 
line  Y'A2Y,  which  makes,  with  OX,  the  axis  of  reference  for  the 
E.M.F's.,  an  angle  equal  to  7-,  and  having  A2Y'  for  its  positive  direc- 
tion. We  thus  obtain  the  diagram  Fig.  22. 

This  diagram  is  bipolar  because  the  axis  to  which  the  E.M.F.  E2  is 
referred  is  the  horizontal  line  OX,  and  the  vector  E\  turns  around  O 


FIG.  22. 

as  a  center;  whereas  the  current  vector  AiA2  turns  around  A2  and  is 
referred  to  the  axis  YA2Yf. 

The  diagram  of  Fig.  22  shows  that  the  current  is  more  or  less  directly 
opposed  to  the  E.M.F.  E2.  It  can  be  decomposed  into  two  compo- 
nents, one  directly  opposed,  the  other  having  a  phase-difference  of  — . 

This  decomposition  is  easily  made  on  the  diagram  (Fig.  23)  by  pro- 
jecting the  vector  A2Ai  on  the  line  A2Y  and  on  a  line  (AiD)  perpen- 
dicular thereto.  We  thus  obtain: 

(i)  The  component  having  the  same  phase  as  E2  which,  following 
the  expression  of  Dolivo-Dobrowolsky,  we  will  call  the  "  watted  " 
current, 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  35 

(2)  The  component  having  a  phase-difference  of  — , 

Id=AlD. 

The  arrows  indicate  the  direction  of  the  vectors.  Taking  into 
consideration  the  direction  of  YY'  (in  the  case  of  Fig.  22),  it  is  seen 
that  every  wattless  current  AiD,  drawn  to  the  left  of  the  line  of  ref- 
erence A2Y,  is  a  current  which  lags  with  respect  to  the  E.M.F.  E2  but 


which  leads  with  respect  to  the  E.M.F.  E\.  If  the  point  A\  should 
come  to  the  right  of  A%Y\  (Fig.  23),  the  wattless  current  would  lead 
with  respect  to  E2  but  would  lag  with  respect  to  E\. 

Applications  of  the  Diagram  of  the  First  Kind.  The  diagram  has 
for  its  object  to  enable  the  different  working  conditions  of  the  motor 
to  be  grasped  at  a  glance.  It  lends  itself  especially  well  to  this 
when  the  motor  E.M.F.  E2  is  maintained  constant,  because  only 
the  point  A\  is  then  displaced,  as  the  result  of  a  variation  either  of  the 
phase-angle  6,  or  else  of  the  generator  E.M.F.  E\.  In  both  cases 
the  outputs  are  denned  by  the  curve  which  AI  describes  and  which  we 
will  call  the  "locus"  of  AI.  This  locus  is  a  circle  when  the  excita- 


36  SYNCHRONOUS  MOTORS 

tion  of  the  generator  is  also  constant.  Each  position  of  the  point  A\ 
defines  the  vector  of  the  E.M.F.  Ei  =  OAi  and  the  vector  of  the  cur- 
rent ZI  =  A2Ai.  Knowledge  of  any  two  whatever  of  the  elements 
E^  6,  I,  (f>  leads  to  that  of  the  two  others. 

The  diagram  can  be  employed  when  E2  varies,  but  the  point  O  then 
varies  at  the  same  time  as  A\,  so  that  the  locus  of  A\  no  longer  suffices 
to  define  the  conditions  of  operation.  There  is  then  some  advantage, 
if  EI  is  constant,  in  having  recourse  to  the  second  kind  of  diagram. 

Line  of  Equal  Power  Occurring  with  Constant  Excitation.  The 
electric  power  of  the  motor,  being  equal  to  the  product  of  the  E.M.F. 
E2  by  the  watted  current  Iw,  will,  when  the  excitation  is  constant,  be 
proportional  to  the  segment  A2D.  When  the  motor-excitation  remains 
constant,  all  outputs  corresponding  to  equal  watted  currents  are  repre- 
sented by  points,  A\,  situated  on  a  single  right  line  A\D,  perpendicular 
to  the  axis  A2Y.  This  right  line  A^D  is,  therefore,  a  line  of  constant 
power  for  the  motor,  whatever  may  be  the  E.M.F.  of  the  generator, 
which  may  be  variable.  The  same  thing  is  true  for  all  parallel  right 
lines.  It  is  possible  to  represent,  on  the  diagram,  a  series  of  such  right 
lines,  each  bearing  the  indication  of  the  corresponding  power  E2IW 
under  constant  excitation;  and  then,  for  each  position  of  the  point 
AI,  the  corresponding  power  will  be  known. 

The  mechanical  power  is  equal  to  the  electric  power  less  the 
losses  due  to  eddy  currents  and  hysteresis.  (The  heat-losses  are 
not  included  in  the  electric  power  E2IW).  The  effect  of  these  losses 
is  itself  equivalent  to  a  certain  appreciable  watted  current,  y0,  and 
to  a  wattless  current  which  can  be  practically  neglected.  There- 
fore it  will  only  be  necessary  to  draw  on  A2Y  a  vector,  A2d=Zjo, 
which  represents  the  "  lost  "  current;  and  the  power  that  is  really 
useful  will  be  proportional  to  the  segment  dD.  Although  the  loss 
under  consideration  varies  slightly  with  the  output,  it  can,  without 
material  error  in  constructing  the  diagram,  be  considered  as  constant 
for  a  given  excitation,  and  as  equal  to  the  power  necessary  to  operate 
the  motor  without  load.  The  electric  power  supplied  by  the  generator 
is  deduced  immediately  from  the  electric  power  of  the  motor  by  adding 
the  resistance-losses  (I2R)  in  the  circuit. 

Lines  of  Equal  Phase.  Since  the  phase  (j>  of  the  current  is  measured 
by  the  angle  AiA2D,  all  the  points  representing  outputs  AI,  located 
on  the  same  right  line  issuing  from  A2,  correspond  to  the  same  angle 
of  lag  <f>.  Any  line  issuing  from  A2  therefore  constitutes  a  line  of  equal 
phase. 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  37 

Limit-Circle  of  Current.  Let  Im  be  the  maximum  value  of  the  cur- 
rent which  the  armature  can  withstand.  If,  around  A  2  as  center,  a 
circle  be  drawn  with  a  radius  equal  to  Im  according  to  the  scale  of 
amperes  (or  equal  to  ZIm  according  to  the  scale  of  volts),  this  circle 
will  constitute  the  boundary  of  a  space  having  certain  characteristics. 
All  load-points  inside  that  space  correspond  to  loads  which  can  be 
maintained  indefinitely  without  the  current,  7,  exceeding  that  limit. 
All  load-points  situated  outside  this  space  represent  loads  which  can- 
not be  maintained  indefinitely.  This  limit-circle  of  current  is,  therefore, 
also  a  limit-circle  of  stability  of  operation,  of  the  motor. 

Algebraical  Relations  Deduced  from  the  Diagram.  To  facilitate 
graphical  calculation  it  is  always  useful  to  have  the  algebraical  expres- 
sions for  the  variables.  What  we  are  interested  in  knowing  is  the  cur- 
rent strength,  its  phase-angle,  the  power  of  the  motor  as  a  function  of 
the  E.M.F.'s  E2  and  E\,  of  the  constants  of  the  circuit,  and  of  the 
angle  of  lag  6,  or  conversely,  any  of  these  as  a  function  of  the  power,  etc. 

The  solution  of  the  triangles  represented  in  Fig.  23  gives  imme- 
diately the  relation  sought.  In  the  triangle  AiDA2  we  have 

Id=I  sin  <£,     ........     (i) 

Iw  =  Ico$(f)  .........     (2) 

Again,  the  projection  ZI   is   equal    to  the    projection   of  the  broken 
line  AiDA2: 

ZI  cos  (r-  <£)  =  Zld  cos  *[  +  Zld  sin  f       .......     (3) 

=  RIw+XId. 

The  triangle  OA2Ai  enables  EI  or  /  to  be  expressed  in  terms 
of  other  quantities,  thus: 


Icos(r-<l>),  .     ...     (4) 
(I)2=E12+E22-2E2ElcosO  .......     (5) 

Finally,  if  the  triangle  OA2Ai  be  projected  on  the  line  A2D  and 
on-  a  line  DAl  perpendicular  thereto,  we  will  have  Iw  and  Id  as  a  func- 
tion of  EI,  E2  and  0: 

ZIW  =  E1  cos  (r-0)-E2  cos  r,   .     ,     ....     (6) 

smr.      .     .     ,     .     .     .     (7) 


38  SYNCHRONOUS  MOTORS 

The  power  of  the  motor  and  that  of  the  generator  will  be  easily 
obtained  by  noting  that  the  phase-angle,  which  is  <p  with  respect  to  EI, 
becomes  (j)+6.  We  will  therefore  have 


(8) 

.....     (9) 
or,  replacing  7  cos  (f>  and  /  sin  </>  by  their  values,  Iw  and  Id,  we  have 


p  _E\  \     cos6[Ei  cos  (f—6}  —  E2  cos  y] 

1       ~~v" 


(10) 

r\ 

-sin  6[Ei  sin  (j—0)-E2  sin 
which  can  also  be  written 

77, 

(n) 


Numerical  Example.  Let  us  consider  the  case  of  a  power-trans- 
mission to  a  distance  of  10  kilometers  by  two  Mordey  alternators  of  37 
kilovolt-amperes  (2000  volts  and  18.5  amperes),  having  a  frequency 
of  100  cycles,  the  resistance  being  3  ohms  and  the  reactance  43  ohms.1 
Let  us  assume  that  the  loss-allowance  in  the  line  is  200  volts  at 
20  amperes.  The  line  will  therefore  have  a  resistance  r'  =  io  ohms. 
If  the  conductors  have  a  resistivity  do)  of  2  microhms  per  centimeter- 
cube,  the  sectional  area  of  each  conductor  is 

2pl     2Xio~6X2oXio5 
$=-5-=:—  —  =0.40  sq.cm. 

r'  10 

=0.062  sq.in., 

and  the  corresponding  diameter  will  be  7.15  mm.  (0.28  inch).  Let  us 
assume  that  the  distance  between  the  wires  is  60  cm.  (23.6  inches). 
The  tables  for  line-inductances  2  give,  for  the  linear  inductance,  in 
millihenrys  per  kilometer,  the  following  values: 

2(0.8188+0.1173)  =  1.872. 

1  This  reactance  was  determined  directly  from  the  characteristic  curves  given 
for  these  machines  by  Mr.  Mordey.     The  value  usually  given  for  the  inductance 
of  this  machine  is  too  low. 

2  See  an  article  by  the  author  in  V  Eclair  age  Electrique,  Oct.-Dec.,  1894. 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  39 

For  a  line  of   10  kilometers   and  a  frequency  of   100  cycles,  the 
reactance  (assuming  the  capacity  to  be  negligible)  will  therefore  be 


10X628X1.872 


IOOO 


=11.8  ohms, 


or  approximately  12  ohms. 

Under  these  conditions,  the  constants  of  the  circuit  are,  in  round 
numbers,  as  follows: 

R  =  i6  ohms,  #=98  ohms;   Z  =  ioo  ohms;   and  we  have 


q8 
tan  r=^-=6. 

'      16 


16 

cosr  =  -  =0.16. 
'      100 


The  ratio  of  the  two  scales  will  be  100.  Therefore  a  distance 
corresponding  to  2000  volts  on  one  scale  will  correspond  to  20  amperes 
on  the  other  scale. 

Suppose,  now,  the  E.M.F.  of  the  motor  to  be  £2  =  2000  volts,  and 
let  it  be  represented  by  the  stationary  vector  A2O  (Fig.  24)  drawn 
on  the  axis  of  OX  in  the  negative  direction.  The  right  line  YA2Yf, 
making  the  angle  f  with  OX,  backward,  will  be  the  origin  of  the 
vectors  representing  currents.  The  lines  of  equal  electric  power  for  the 


40  SYNCHRONOUS  MOTORS 

motor  (not  including  resistance-losses  in  the  armature)  will  be  right 
lines  perpendicular  to  YA2Yf.  In  the  diagram,  several  of  these  right 
lines  have  been  drawn,  corresponding  to  powers  increasing  each  time 
by  5  k.w.  Let  us  complete  the  diagram  by  describing,  from  A  2  as  a 
center,  with  the  radius  ZI  =  100X18.5  A,  the  circle  of  normal  current. 
The  point  AI,  which  depends  on  the  load,  can  be  displaced  anywhere 
in  the  whole  space  inside  the  circle  of  normal  current-value;  but  it 
cannot  be  displaced  outside  this  space,  except  for  a  short  time,  owing 
to  the  heating  of  the  machine.  For  each  position  of  the  load-point  A\, 
its  distance  from  the  point  O,  measured  by  the  scale  of  volts,  represents 
the  corresponding  E.M.F.  of  the  generator,  and  the  distance  AiA2, 
divided  by  Z,  represents  the  value  of  the  corresponding  current. 

If  we  suppose  the  generator  to  be  exactly  like  the  motor,  i.e.,  if  its 
E.M.F.  is  2000  volts,  the  diagram  shows  immediately  that  the  maximum 
power  practicably  obtainable  cannot  exceed  30  k.w.  To  attain  37.5 
k.w.  without  exceeding  the  normal  current,  it  would  be  necessary  to 
raise  the  E.M.F.  of  the  generator,  E\,  to  2960  volts. 

Diagram  of  the    Second  Kind.     The  Vector   of  the  Generator 

E.M.F.,  Ei,  as  a  Fixed  Axis. 
Let  the  vectors  corresponding  to 
the  generator  and  motor  E.M.F.'s 
be  represented  by  two  right  lines, 
OA  i  and  OA2,  respectively  (both 
being  taken  with  the  signs  they 
present  when  the  closed  circuit  is 
followed  around).  The  angle 
AiOA2  (Fig.  25),  which  we  will 
FlG-  25-  still  call  6,  now  measures  the  lag 

of  E2  behind  EI. 

The  current  /  is  equal  to  the  resultant  E.M.F.  A  \A2  divided  by 
the  impedance, 


<»>• 


and  it  lags  by  the  angle  7-  behind  AiA2.  Its  vector  AiC=RI  is  there- 
fore the  projection  of  AiA2  over  that  angle,  and  the  phase-difference 
(p  between  the  current  and  the  active  E.M.F.  Ei  is  the  angle  OA\C. 
In  the  case  shown  in  the  diagram,  the  current  lags  behind  AiO,  i.e., 
behind  the  E.M.F.  EI  measured  at  the  terminals  of  the  generator. 
The  E.M.F.  OAi  can  be  decomposed  into  two  components,  of 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  41 


which  one,  A\H,  is  in  phase,  and  the  other,  OH,  is  in  quadrature, 
with  the  current  /.  The  effect  of  the  motor  is  therefore  equivalent 
to  an  apparent  reactance,  which  is  here .  positive, 


Y         

-*  app.  — 


OH 
A2C 


combined  with  an  apparent  resistance 


/?    -? 

***~~  A& 

Finally,  the  electric  power  developed  by  each  of  the  two  E.M.F.'s 
and  E2  will  be 


R 


and 


P2=E27cos(ZE2,  /) 


•     •     .     (13) 


and 


where  (Z-Ei,  /)  and  (£E2,  I)  mean  the  angle  of  lag  between 
/,  and  between  E2  and  /,  respectively. 

The  ohmic  loss  in  the  circuit  (I2R)  is  equal  to  the  difference 
P\  —  P2.  Let  us  now  transform  the  diagram,  as  above,  in  such 
manner  that  A  i  A  2  shall  represent  the  current,  in  magnitude  and  in 
phase.  To  obtain  the  magnitude, 
it  will  be  sufficient  to  use,  for 
measuring  the  amperes,  a  scale 
Z  times  greater  than  the  scale  of 
volts,  Z  being  the  numerical 
value  of  the  impedance,  expressed 
in  ohms. 

To  obtain  the  lag  of  the  cur- 
rent thus  represented,  with  re- 
spect to  the  generator  E.M.F. 
EI,  it  will  be  sufficient  to  draw 
(Fig.  26)  a  stationary  right  line 
AiN  making  the  angle  y  in  ad- 
vance of  OA\.  It  will  be  seen 
immediately,  on  comparing  Figs. 
25  and  26,  that  the  angle  NA\A% 
formed  by  the  vector  AiA2,  with  the  line  NA±  is  equal  to  the  angle 
OA\C  previously  referred  to,  and,  consequently,  represents  the  apparent 
phase-angle  iff. 


for  current,  I 


FIG.  26. 


42  SYNCHRONOUS  MOTORS 

[To  make  the  diagram  complete  we  should  draw  two  similar  diagrams,  turned 
in  different  directions,  one  corresponding  to  the  volts,  the  other  to  the  amperes, 
each  with  its  own  special  scale.  However,  this  would  be  of  no  advantage,  and 
we  will,  therefore,  retain  a  single  diagram  with  the  two  scales,  remembering  that 
the  origin  of  phases  is  OAl  for  the  E.M.F.'s,  and  A^N  for  the  currents.  We  shall 
indicate,  on  that  diagram,  the  expressions  for  the  segments  of  lines,  as  if  they 
were  all  E.M.F.'s;  and  it  will  be  sufficient  to  divide  them  all  by  Z  to  have  their 
expressions  for  currents. 

For  example,  the  segments  OAt  and  OA2  will  represent,  respectively,  by  the 
scale  of  amperes,  the  two  currents 


(.4) 

'•-7 

which  each  one  of  the  alternators  would  produce,  respectively,  in  the  circuit, 
if  the  other  were  stopped.] 

The  line  A\N  will  therefore  be  the  origin  of  phases  for  the  current 
with  respect  to  the  E.M.F.  EI. 

If  we  project  A\A%,  on  the  line  A\N  and  on  the  line  AA2  per- 
pendicular thereto,  we  will  obtain,  by  the  scale  of  amperes,  two 
current-components  : 

(a)  The  watted  current  i=AiA,  or  the  current  which  is  in  phase 
with  the  E.M.F. 

(b)  The  wattless  current  j  =  AzA,  cr  the  current  which  is  in  quad- 
rature with  the  E.M.F.1 

The  phases  of  the  two  currents  are  both  referred  to  AiN  as  origin. 

The  work  of  Dobrowolsky  has  shown  that  this  decomposition  of  the 
current  is  very  interesting  from  a  practical  standpoint;  it  is  more  so, 
in  general,  than  the  decomposition  of  the  E.M.F.  already  indicated 
(page  41). 

The  power  furnished  by  the  active  E.M.F.  may  be  deduced  from 
the  watted  current,  thus: 


The  power  actually  utilized  may  be  deduced  from  this,  in  the 
following  manner: 


in  which  7*2  =  the  armature-resistance  of  the  motor. 

1  These  terms,  "watted"  and  "wattless,"  are  adopted  here  to  conform  to  the 
international  language  of  manufacturers,  although  they  are  objectionable. 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  43 

Power-Values  as  Function  of  the  Lag-Angle  6.  To  express  the 
power-values  PI  and  P2  in  terms  of  the  angle  0,  it  is  only  necessary 
to  substitute,  in  the  two  equations  (13),  other  values  for  /  cos(ZEi,  /) 
and  for  /  cos  (£E2,  /). 

1  cos  (£Ei,  I)  is  nothing  more  than  the  watted  current  i  for  the 
case  represented  in  Fig.  23. 

Let  Ji  =  ~£    and  J^  =  ~l'    (See]eq.  14,  P-  42-) 

The  diagram  gives,  readily, 

i=I  cos  (ZEi,  7)=/i  cos  r-J2  cos  (0+r).      .     (15) 

The  current  7  cos  (  Z  E2,  I)  will  be  obtained  by  analogy,  by  project- 
ing the  diagram  of  Fig.  26  upon  a  right  line  making  with  OA2  an 
angle  that  is  again  equal  to  the  angle  f.  From  this  we  obtain  the 
absolute  value 

I  cos  (Z.E2,  I)=Ji  cos  (-]r—d)  —  J2  cosy  .....     (16) 

77"       77 

Replacing  J\,  J2  by  their  values,  ~,  —  ,  we  will  therefore  have 

P1=[£1cosr-E2cos(0-r)]  .....     (17) 


P2  =      [E2  cos  (r  -  0)  -  E2  cos  r]  .....     (18) 

These  are  expressions  wherein  only  two  variables,  E2  and  6,  enter. 
The  utility  of  these  expressions  will  be  shown  later. 

Use  of  this  Diagram  for  the  Study  of  Different  Loads.  These  loads 
cannot  be  determined  without  imposing  some  condition  which  makes 
the  variables  (i.e.  E.M.F.  current,  power,  lag,  etc.,  )vary  according  to 
some  definite  law.  In  that  case,  to  each  value  of  the  lag  there  will 
correspond  a  single  (or  several)  quite  definite  values  of  the  E.M.F.  E2) 
and  vice  versa.  Consequently,  on  the  diagram,  the  point  A2,  instead 
of  being  able  to  range  over  the  whole  surface,  can  be  displaced  only 
on  a  single  curve  such  as  CC',  for  example  (Fig.  26),  whose  equation, 
in  polar  co-ordinates,  with  respect  to  the  point  O  as  a  pole,  is  noth- 
ing more  than  the  relation  imposed  between  E2  and  6. 

For  each  point  of  this  curve,  which  may  be  called  the  locus  of  A2, 
there  correspond  two  radial  vectors  issuing  from  two  fixed  poles. 

One  of  these  vectors,  OA2,  is  drawn  from  the  point  O  as  a  pole; 
and  it  indicates  the  E.M.F.  E2,  in  magnitude  and  in  phase  (starting 
from  OAi)]  the  other,  A^A2,  is  drawn  from  the  point  AI  as  a  pole; 


44 


SYNCHRONOUS  MOTORS 


and  it  indicates  the  corresponding  value  of  the  current,  in  magnitude 
and  in  phase  (starting  from  ^lAT"). 

Knowledge  of  any  two  of  the  elements  E2,  0,  I,  enables  all  the 
others  to  be  known,  and  also  the  lag  0  and  the  reactive  and  active 
currents. 

When  EI  is  variable,  this  diagram  becomes  inconvenient,  and 
preference  should  be  given  to  the  diagram  of  the  first  kind. 

Curves  of  Constant  Electric  Power  of  the  Motor  when  the 
Generator  has  Constant  Excitation.  These  curves,  which  indicate 


FIG.  27. 

the  p&wer  developed  by  the  motor  for  each  position  of  A2,  in  the  diagram, 
are  represented  by  Eq.  (18),  in  polar  co-ordinates  referred  to  the 
pole  O  (Fig.  27)  (P2  being  supposed  constant),  and  they  now  become 
circles  instead  of  straight  lines.  Let  us  take  two  rectangular  axes 
passing  through  O,  the  axis  of  x  being  directed  along  the  line  ON, 
which  makes  with  OAi  the  angle  f.  We  can  then  write 

£2cos  (r-0)=x, 


f£2co 
(E22= 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  45 

By  substitution,  we  have 


cos  f        cos^  7- 
or 


2  cos  TY  cos^  f 

from  which  it  can  be  seen  that  the  curve  is  a  circle  whose  center  has 
the  following  co-ordinates: 

El 

y=o. 

Each  circle  has  for  its  center  the  point  of  intersection,  N,  of  the 
right  lines  ON  and  OA^,  both  making  the  angle  7-  with  OAi\  and  the 
radius  of  this  circle  is 


This  result  is  also  easily  obtained  by  geometry. 

As  for  the  lines  of  constant  electric-power,  PI  (applied  at  the 
terminals),  they  are  evidently  straight  lines  perpendicular  to  AiN  (none 
of  which  has  been  drawn  on  the  diagram  except  the  line  PI=O); 
because  they  are  denned  by  the  simple  condition  that  the  current 
which  is  active  with  respect  to  E\,  (i.e.,  i),  is  constant. 

The  expression  (19)  can  be  calculated  algebraically,  for  various 
values  of  P2,  or  it  can  even  be  determined  graphically  by  means  of 
rectangular  triangles.  It  is  convenient  to  express  the  power  P2  as  a 
function  of  the  maximum  power  which  can  be  obtained  by  making 
E2  vary,  E\  being  supposed  to  remain  constant.  As  will  be  seen  later 
this  power,  which  will  be  designated  by  the  letter  P,  has  the  following 
value: 

p=f :  •  •  •  •  <»> 

It  is  easy  to  construct  a  series  of  equipotential  lines  corresponding  to 
the  given  values  of  P2,  and   thus   to  predetermine   the  variations  of 
power  resulting  from  the  displacements  of  the  point  A^. 
The  radius  p  can  then  be  written, 

'     El 


46  SYNCHRONOUS  MOTORS 

It  is  therefore  sufficient  to  divide  the  segment  A±N  in  the  proportion 
of  Y1 — p"  to  obtain  the  power-lines  shown  in  Fig.  27. 

If,  for  example,  we  take  the  series  of  values, 
P2 

-p-  =  O.IO,    O.2O,    0.30,    0.40,    0.50,    O.6O,    O.yo,    O.8o,    O.gO,    I.OO, 

we  will  find 

-^—=0.949,  0.894,  0.837,  o-774,  0.707,  0.632,  0.548,  0.447,  0.316. 

In  practice,  these  lines  should  be  graduated  in  kilowatts,  in  the 
usual  way. 

It  is  often  unnecessary  to  draw  the  lines  for  the  high  powers,  since 
the  power  P  is  much  greater  than  that  which  the  armature  can  with- 
stand without  dangerous  heating,  in  continuous  operation. 

Since  cos  7-  is  very  small  (usually  <o.io),  in  ordinary  practical 
cases,  the  center  N  of  the  circles  will  often  be  outside  the  diagram,  but 
it  will  be  always  easy  to  draw  the  circles  by  points,  from  their  equation 
in  rectangular  co-ordinates. 

Moreover,  in  that  case,  the  circles  are  of  very  large  radius,  and  are 
quite  sufficiently  determined  by  two  tangents.  Now  for  each  circle, 
such  as  GFG',  the  directions  of  the  two  tangents  at  F  and  G'  are  known, 
because  they  are,  respectively,  perpendicular  to  A\N  and  ON.  Again, 
the  position  of  the  points  F  and  G'  corresponding  to  a  power  PI,  is 
given  by  the  obvious  equation  : 


It  is  very  easy,  therefore,  to  construct  these  two  tangents  and  then 
to  draw  approximately  the  circular  arc  which  they  determine. 

It  will  be  noted  that  the  "  P%"  power-values,  in  Fig.  27,  are  positive 
above  the  point  A\,  and  negative  below  it.  This  shows  that  the  alter- 
nator will  operate  as  a  motor  in  a  certain  portion  of  the  plane,  contained 
inside  the  power-circle  ^2=0,  and  that  it  will  operate  as  a  generator 
beyond  that  circle.  In  the  cross-hatched  space  comprised  between  the 
lines  P2=^o  and  PI=O  it  cannot  operate  either  in  one  way  or  in  the 
other,  because  the  current  which  would  then  pass  through  the  machine 
would  cause  a  loss  of  energy  in  the  armature  greater  than  the  available 
power. 


DETAILED  STUDY  OF  OPEEATION  WITH  NOKMAL  LOAD  47 

Current-limit  Circle.  It  is  known  that  the  armature  cannot  indef- 
initely withstand  a  current  exceeding  a  certain  limiting  value  Im.  If 
we  draw,  around  the  pole  AI  as  a  center,  a  circle  of  radius  precisely 
equal  to  Im  (by  the  scale  of  amperes),  all  the  points  A2  situated  out- 
side that  circle  will  give  values  I=AiA2  >/w;  whereas,  inside  that  circle 
the  value  of  7  will  be  <  Im.  The  circle  drawn  is  therefore  a  limit 
beyond  which  the  load -point  must  not  go,  for  continuous  operation. 

Lines  of  Equal  Phase.  In  the  same  way  that  lines  of,£jqual  power 
have  been  drawn,  lines  of  equal  current-phase  can  be  easily  drawn, 
such  that,  at  the  loads  represented  by  their  various  points,  the  phase- 
angle  between  the  E.M.F.  and  the  current  will  always  be  the  same. 

If  the  phase  of  the  current  /  is  measured  from  the  E.M.F.  E^ 
these  lines  are  evidently  straight  lines  issuing  from  the  point  A\.  The 
right  line  AiN  corresponds  to  "zero"  phase-angle.  All  the  loads  to 
the  right  thereof  correspond  to  a  lag  in  phase,  and  all  the  loads  to  the 
left  correspond  to  a  lead  in  phase.  (Fig.  27.) 

To  obtain  lines  corresponding  to  currents  of  equal  phase  with 
respect  to  the  E.M.F.  E2,  we  note  that  all  the  points  for  any  current 
value  whatever,  among  these  lines,  should  present  a  constant  angle 
OA2A\.  These  lines  are  therefore  circles  having  OA\  as  a  chord. 
If,  for  example,  we  wish  the  line  of  phase  corresponding  to  the  lag 
a  between  E2  and  /,  we  construct,  on  OAl  as  a  chord,  a  circle  admitting 
of  the  angle  a—  f  (since  the  true  phase  of  the  current  lags  by  the  angle  7- 
with  respect  to  the  line  AiA2). 

Only  the  circle  corresponding  to  an  angle  a=n  has  been  drawn, 
in  the  example  represented  in  the  diagram.  This  circle  represents 
the  exact  opposition  in  phase  between  the  current  and  the  internal 
E.M.F.  E2.  It  is  easily  seen  that  this  circle  is  tangent  to  the  two 
lines  ON  and  Aj^N. 

For  all  loads  corresponding  to  points  situated  inside  this  circle, 
the  phase  is  >?r,  outside  the  circle,  on  the  contrary,  it  is  <  TT. 

[It  will  be  noted,  in  passing,  that  the  circle  of  zero  power  P2=o, 
passing  through  the  point  AI  is  nothing  more  than  the  circle  of  phase 

— ,  because  the  power  is  zero  every  time  that  the  current  is  in  quad- 
rature with  the  E.M.F.] 

Numerical  Example.  Let  us  return  to  the  example  of  power-trans- 
mission by  means  of  two  Mordey  alternators,  for  which  the  numerical 
data  have  already  been  given  (page  38),  and  let  us  apply  the  second 
diagram  to  this  case. 


48 


SYNCHRONOUS  MOTORS 


The  maximum  theoretical  power  is 

2 


-^        I    /2OOO\2 

P=— I-    -)  =62,500  watts. 


The  radii  of  the  power-circles  are  given  by  eq.  (21);   whence  we 
have 


'  =  6250^1—16 


iol 


in  which  P2  is  expressed  in  watts,  p  being  measured  by  the  scale  of 
volts.     In  this  manner  we  find  the  radii  for  the  circles 


of 
to  be 


and 
and 


50  k.w. 
2800  volts. 


5        .     10  20  30  40 

6000,        5700,       5130,       4520>       375°> 

These  simple  calculations  give  all  the  elements  necessary  for  the 
diagram  in    Fig.   28.      In  this  case  the  current-circle  corresponding  to 


-58S5- 


FIG.  28. 

the  normal  current  of  18.5  amperes  has  also  been  drawn,  in  addition, 
to  show  the  limit  of  loads.  Let  A  2  be  any  position  whatever  of  the 
load-point,  within  this  circle;  it  is  only  necessary  to  measure  the  various 
lines,  according  to  the  proper  scales,  to  see  that  the  corresponding 
voltage  E2  is  3000  volts,  the  current  strength  is  n  amperes,  the 
reactive  current  (leading)  is  10.5  amperes,  and  the  power  is  5  k.w.  The 


DETAILED   STUDY  OF  OPERATION  WITH  NORMAL  LOAD  49 

minimum  current  corresponding  to  the  same  power  would  be  OD  = 
2.8  amperes,  approximately. 


II.    OPERATION    OF   A   MOTOR   WITH    CONSTANT    EXCITATION, 
SUPPLIED   AT   CONSTANT   E.M.F. 

The  two  diagrams  lend  themselves  equally  well  to  the  discussion 
of  this  case,  which  is  usually  the  most  frequent  one  in  practice.  In 
the  first  diagram  (Fig.  29)  the  E.M.F.  vector  of  the  motor  will  remain 
stationary  and  that  of  the  generator  will  turn  round  the  pole  O,  when 


FIG.  29. 

the  load  varies.  The  extremity  A\  describes  a  circle  having  O  for  its 
center  and  EI  for  its  radius.  If  the  lines  of  equal  load  have  been  drawn, 
their  points  of  intersection  with  the  circle  will  define  completely  the 
conditions  (currents  and  lags)  for  each  load.  The  useful  active  cur- 
rents will  be  obtained  by  subtracting  the  loss-current  —~  from  the 

active  currents  themselves. 

It  will  be  noted  that  the  power  is  zero  when  A\  is  on  the  power 
line  passing  through  d.     If,  in  consequence  of  overload,  the  motor- 


50  SYNCHRONOUS  MOTORS 

speed  diminishes,  6  increases  and  the  power  developed  also  increases. 
The  condition  is  therefore  stable  for  low  powers. 

But  the  power-output  is  limited.  Its  maximum  corresponds  to 
the  power-line  drawn  tangentially  to  the  circle.  The  correspond- 
ing output,  N,  is  the  limit  of  possible  outputs,  because  the  effect  of 
an  overload,  by  increasing  further  the  lag  6,  would  be  to  bring  the  vector 
OA  i  into  a  position  such  as  OM,  where  the  power-output  would  be 
diminished,  and,  consequently,  the  motor  would  stop. 

The  line  ON,  drawn  through  O  parallel  to  A2Y,  therefore  defines 
theoretically  the  limiting  line  of  stability;  and  the  limiting  lag  between 
the  E.M.F.'s.  is  equal  to  f.  The  maximum  electric  power  applied 
to  the  motor  is  equal  to  the  product  of  the  E.M.F.  E2  by  the  corre- 

A2N' 
sponding  active  current  — — . 

Zj 

Likewise,  on  the  second  diagram  (Fig.  27),  if  the  excitation  is  sup- 
posed to  be  constant  and  if  the  load  is  made  to  vary,  the  end  of  the 
E.M.F.  vector  E2  describes  a  circle  around  O  as  a  center.  Its  point  of 
intersection,  A  0,  with  the  circle  of  zero-power  (P2=^o),  gives,  theoret- 
ically, the  condition  of  operation  without  load;  but,  practically,  the 
power  necessary  to  overcome  friction  and  other  losses  not  being  equal 
to  zero,  the  unloaded  condition  will  correspond  to  a  higher  point, 
such  as  A"Q,  situated  on  the  power-circle  P2=o.io  P. 

When  the  load  applied  to  the  motor-shaft  increases,  the  point  of 
output  A2  is  displaced  on  its  circle  in  the  direction  of  increasing  powrer- 
outputs,  i.e.,  toward  the  top,  and  the  angle  of  lag  6  and  the  current 
/  increase  correspondingly. 

The  power  can  increase  up  to  the  point  Q2,  where  the  characteristic 
is  tangent  to  a  power-circle  (Fig.  27);  but  it  could  only  decrease 
if  the  angle  of  lag  were  to  be  further  increased.  The  direction  ON 
therefore  marks  the  theoretical  limit  of  stability  of  operation.  When 
the  angle  of  lag  is  increased  further  the  motor  falls  out  of  step  and  comes 
to  a  stop. 

It  may  be  seen  in  the  same  manner  that  the  alternator  operates 
as  a  generator  below  the  point  A  '0,  and  that  the  necessity  of  stability 
of  operation  limits  the  angle  of  lag  to  the  direction  OM,  which  is  per- 
pendicular to  the  right  lines  of  equal  power  PI,  and  symmetrical  with 
ON,  with  respect  to  OAi. 

Maximum  Power.  The  maximum  power  P2,  corresponding  to 
the  limit  of  stability  just  determined,  can  be  easily  calculated  by  mak- 
ing 6  =  ?  in  the  general  power  equation  (18). 


DETAILED   STUDY  OF  OPERATION  WITH   NORMAL  LOAD  51 

From  this  we  get 


COS  r  (Ei-  E2  COS 

=E2-        —  - 


The  power-value  will  be  higher  in  proportion  as  the  impedance  Z  and 

the  resistance  R  are  smaller  (  provided  that  cos  r  <-  -=^  ). 

\  2  A2/ 

The  formula  (22),  can  also  be  written  directly  from  the  diagram 
(Fig.  29)  corresponding  to  the  load  which  is  at  the  limit  of  stability, 
because,  we  have,  from  this  diagram, 

ZIW  =Ei  —  E2  cos  f, 
and  also  P2=E2Iw. 

In  well-constructed  modern  motors  (having  low  resistance  and  low 
impedance),  there  is  usually  no  tendency  of  the  machine  to  fall 
out  of  step  from  gradual  overloading,  when  it  is  supplied  from  a  con- 
stant potential  circuit,  because,  as  Fig.  28  shows  clearly,  the  current 
becomes  excessive  before  the  power-outputs  approach  the  limit  of 
stability  just  indicated.  For  example,  the  Labour  synchronous  motors, 
which  can  be  cited  as  excellent  existing  examples  of  this  class  of  appa- 
ratus, are  only  limited,  in  regard  to  load,  by  the  heating  of  the  armature, 
and  not  by  the  tendency  to  fall  out  of  synchronism.  It  is  the  circle  of 
maximum  current,  drawn  as  we  have  already  seen,  which  limits  long- 
continued  loads. 

But  loss  of  synchronism  may  result  from  anything  which  can 
increase  the  impedance  and  diminish  the  reactance-factor,  and,  con- 
sequently the  tendency  to  fall  out  of  step  is  increased  by  the  resistance 
of  the  line  in  long-distance  transmission. 

It  is  therefore  necessary  to  be  able  to  foresee,  under  certain  cir- 
cumstances, whether  the  operation  will  be  stable  or  not. 

Means  of  Determining  the  Practical  Stability  of  Synchronous 
Motors.  In  practice,  owing  to  oscillations  of  speed,  the  load  is  restricted 
to  a  value  which  is  less  than  the  maximum  power  just  determined. 
It  can  be  conceded,  in  all  cases,  that,  during  these  speed-oscillations, 


52  SYNCHRONOUS  MOTORS 

the  excitation  of  the  motor  (even  if  it  is  compounded)  has  not  time  to 
vary.  It  can,  therefore,  always  be  assumed  that  the  motor  has  a  constant 
induced  E.M.F.,  and  the  results  of  experiments  made  with  such 
motors  can  always  be  used  as  the  starting  point. 

On  this  basis,  the  stability  of  operation  can  be  calculated  in  a 
sufficiently  precise  manner  by  simply  comparing  the  effective  load  of 
the  motor  at  the  maximum  power,  Pm,  which  it  might  develop  with 
its  E.M.F.  E2,  and  the  power,  P2max.,  wm'ch  is  given  by  eq.  (22). 

[This  value  can  be  determined  experimentally  by  the  method  set 
forth  in  the  author's  pamphlet  on  the  "  Coupling  of  Alternators  ", 
Paris,  1894,  p.  4.] 

Experience  shows  that  it  is  possible  to  have  stable  operation  so  long 
as  the  power  P2  does  not  exceed  about  two-thirds  of  Pm,  when  the 
apparatus  driven  by  the  motor  constitutes  a  very  constant  load,  or 
about  the  half  of  Pm,  when  the  load  is  .somewhat  irregular.  [It  is 
assumed,  of  course,  that  the  apparatus  can  withstand  the  corresponding 
current  without  overheating.] 

These  figures  have  been  verified  by  various  experiments  and  they  are 
in  accordance  with  the  results  published  by  other  experimenters.  On 
analyzing  the  power  transmission  at  Cassel  (see  La  Lumiere  Elecirique, 
Vol.  XLV,  p.  616)  where  the  motors  furnish  power  for  charging  bat- 
teries, we  find  that  the  ratio  of  practical  to  theoretical  maximum 

power  is  as  high  as  -—  =  —  ,  but  this  is  exceptional.    It  is  safer  to  assume 
*m      S 

l*=L 

Pm       2* 

Variations  of  Stability  with  Operating  Conditions.  It  is  very 
evident  that,  all  things  being  equal,  the  stability  will  decrease  with 
the  load.  To  see  how  it  varies  with  the  conditions  of  the  circuit 
and  of  the  motor-excitation,  when  the  power  P2  and  the  voltage  E\  are 
constant,  it  is  only  necessary  to  study  the  variations  of  the  power  Pm, 
by  means  of  equations  (22).  With  equal  resistance  it  presents  a 
maximum  of  maximums 


whenever  EI,  E2)  Z,  and  y  satisfy  the  condition 

F         EI        EiZ 

*-  ••••••  (24) 


DETAILED   STUDY  OF  OPERATION  WITH  NORMAL  LOAD  53 

(i)  Suppose  a  given  motor-load,  and  suppose  the  E.M.F.'s.  EI 
and  E2  to  be  nearly  equal.  The  stability  will  first  increase  with  the 
reactance  until  we  have 


i.e.,  until  Z  approaches  27?;   and  it  will  then  decrease  if  —  is  greater. 

K. 

Z 

Inasmuch  as  —  is  always  greater  than  2  in  ordinary  alternators,  any 
K. 

increase  of  inductance  is  objectionable  in  a  motor  which  is  connected 
directly  to  the  source  of  current.  It  could  only  be  useful  if  the  motor 
was  connected  to  the  generator  by  a  Very  long  line  of  exceptionally 
high  resistance. 

(2)  Suppose  a  given  motor-load  and  suppose  the  external  E.M.F. 
EI  to  be  constant.  The  stability  of  operation  will  always  vary  inversely 
with  R-  but  the  result  will  depend  upon  whether  cos  f  be  greater  or 

/  X\ 

smaller  than  -*,  i.e.,  whether  the  reactance-factor  (  tan  r=-^  )  will  be 

_  \  K/ 

higher  or  lower  than  V$.  When  cos  y  >i  an  increase  of  the  internal 
E.M.F.  E2  until  it  exceeds  the  external  voltage  EI  will  cause  an  increase  of 
stability,  whereas,  when  cos  y  <  J  the  increase  will  cause  a  reduction  of 
stability.  This  observation  explains  someapparent  contradictions  of  ex- 
perience. When  a  motor  is  connected  on  a  constant  potential  line  or  on  a 
line  of  low  resistance,  it  is  more  difficult  to  make  it  fall  out  of  step  when  it 
is  over-excited.  If,  on  the  other  hand,  the  same  motor  is  used  for  power- 
transmission  over  a  line  of  high  resistance,  it  will  be  observed  that  increas- 
ing the  excitation  will  also  increase  its  tendency  to  fall  out  of  step. 

In  practice,  the  first  case  is  the  more  frequent,  and  it  is  advisable, 
in  order  to  improve  the  stability,  to  increase  the  excitation  with  the 
load  either  by  hand  or  automatically. 

At  the  same  time,  it  is  seen  that,  in  the  most  frequent  practical  cases, 
it  is  necessary,  in  order  to  have  good  synchronous  motors,  to  diminish 
both  the  resistance  and  the  reactance  of  the  armature.  This  means 
that  the  field  must  be  powerful  and  the  armature-reaction  must  be  low. 

If  we  were  to  make  Z  constant  by  hypothesis,  we  would  be  led 
to  reduce  as  much  as  possible  the  resistance  and  to  increase  the 
reactance;  but  this  case  does  not  occur  in  practice. 

Numerical  Example.  Let  us  take,  for  example,  two  Mordey  alter- 
nators of  37.5  kilowatts  electrical  output  l  coupled  for  electrical  trans- 

1  The  power  available  at  the  shaft  must  be  reduced  in  proportion  to  the 
efficiency  (exclusive  of  the  excitation-energy). 


54  SYNCHRONOUS  MOTORS 

mission,  and  let  £1=2200  volts  be  the  generator  E.M.F.  The  internal 
resistance  of  each  machine  is  3  ohms  and  its  reactance  at  100  cycles 
is  43  ohms,  as  previously  stated. 

(i)  Let  us  first  suppose  the  line  to  have  no  resistance  and  no  react- 
ance. The  total  circuit  will  have  the  following  constants  —  resistance 
6  ohms,  reactance  86  ohms  and  impedance  86.20  ohms. 

The  maximum  power 

P=  —  ;-—  -=201,660  watts, 

4X0 

could  only  be  obtained  by  giving  to  the  motor  E.M.F.  the  value 

E2=—     —  =  15,800  volts, 
0.0695 

and  it  could  only  be  obtained  with  a  current 

T          220°  O 

"  3 


It  is  therefore  unattainable  from  all  points  of  view. 

If   we   limit   the    E.M.F.'s   to    E2  =£1  =  2200   volts,   the   possible 
maximum  will  be 

/  \n 

-(l  -0.0695)  =52,000  Watts; 


i.e.  about  3/2  times  the  normal  output.  The  stability  will  therefore  be 
sufficient  provided  the  machines  are  not  subjected  to  variations  of  load. 

Every  increase  in  voltage  will  here  produce  an  increase  of  (^max* 
If,  for  example,  we  take  £2=3000  volts,  69  k.w.  instead  of  52  is 
found  to  be  the  maximum  output  possible  without  causing  the  machine 
to  fall  out  of  step.  The  stability  is  then  excellent  at  the  normal  load 
of  37.5  k.w.  If,  on  the  contrary,  the  voltage  is  reduced,  so  high  a  load 
could  no  longer  be  attained. 

(2)  Suppose  now  the  resistance  of  a  line  to  be  interposed  between 
the  two  alternators.  Stability  will  evidently  become  impossible  as  soon 
as  the  maximum  power  P  approaches  the  effective  power  P2.  If  we 
take,  for  the  coefficient  of  load,  the  figure  3/2  already  mentioned,  the 
maximum  power  P  should  not  be  less  than  3/2X37.5  =  56  k.w.,  which 
corresponds  to  a  line-resistance  of  about  16  ohms. 

In  his  experiments  Mordey  (Proc.  Inst.  of  Electr.  Eng.,  1894)  was 
able  to  increase  the  resistance  to  very  much  higher  values  (140  ohms), 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  55 

because  he  made  his  machines  run  without  load.  This  proves  nothing, 
however,  in  regard  to  their  operation  under  load.  Moreover,  these 
conditions  were  absolutely  exaggerated,  since  lines  of  such  high  resist- 
ance would  never  be  used  in  practice. 

Q  S- 

Finally,  when  we  attain  a  resistance  of  approximately  R=—-==$o 

<x/3 

ohms,  we  have  cos  <^>=J.  The  maximum  power  for  E2=Ei  being 
then  no  longer  22,550  watts,  the  output  of  the  motor  can  scarcely 
exceed  15  k.w.  Besides,  it  is  sure  that  every  increase  of  the  E.M.F. 
E2  above  2200  volts  will  diminish  the  stability.  This  will  be  all  the 
more  true  if  R  is  higher.  We  can  thus  understand  that  in  using  machines 
of  50  k.w.  with  an  external  or  line  resistance  of  63  ohms,  Mordey  could 
only  operate  these  machines  without  load,  and  that  the  slightest 
increase  of  the  voltage  E2  above  E\  (20  per  cent)  would  cause  the 
machines  to  fall  out  of  step. 

The  preceding  analysis  therefore  explains  perfectly  the  anomalies 
of  Mordey's  experiments. 

We  would  be  brought  to  analogous  conclusions  by  the  considera- 
tion of  sudden  overloads  and  of  oscillating  changes  of  load,  which 
will  be  discussed  later. 

[The  amounts  of  sudden  overloads  allowable  are  discussed  in 
the  author's  preceding  work,  "  The  Coupling  of  Alternators,"  and 
also  in  La  Lumiere  Electriqite,  Vol.  XLV,  p.  474.] 

III.  COMPARISON  OF  POSSIBLE  OUTPUTS  AT  CONSTANT  LOAD 
WITH  VARIOUS  EXCITATIONS.  CONSTANT  POTENTIAL 
SUPPLY 

Existence  of  a  Current-Minimum.  There  exist  several  values  of 
current  and  E.M.F.  which  permit  a  motor  subjected  to  an  external 
constant  E.M.F.  E\  to  develop  a  given  electrical  power  P2.  These 
are  all  values  for  which  the  point  A2  (Fig.  27)  falls  on  the  line  of 
equal  power  corresponding  to  that  value  P2. 

These  values  differ  from  each  other  by  the  values  of  the  E.M.F.  E2 
necessary  to  produce  them,  and  also  by  the  phase-angles  6  and  <£>  of 
the  corresponding  current  /,  and,  particularly,  by  the  reactive  current. 

If,  to  simplify  matters,  we  suppose  the  motor  to  be  supplied  from  a 
constant  potential  source  whose  impedance  is  negligible,  the  diagram 
in  Fig.  27  will  give  the  solution. 

The  current  being  represented  by  the  vector  A^A2  according  to 
the  scale  of  amperes,  its  minimum  evidently  corresponds  to  the  point 


56  SYNCHRONOUS  MOTORS 

F  on  the  power-circle  G  G',  situated  on  the  phase-line  AN.  As  soon 
as  the  point  A2  is  displaced  on  the  circle  to  either  side  of  the  line  AN, 
the  current  increases.  There  is  therefore  a  value  of  the  internal  E.M.F. 
£2  which  corresponds  to  a  minimum  current,  and  all  values  greater 
or  less  than  this  value  necessitate  a  stronger  current. 

It  is  evident  from  Fig.  27,  without  any  explanation,  that  the  minimum 
value  is  that  which  brings  the  current  into  phase-coincidence  with  the 
E.M.F.  EI,  i.e.,  with  the  E.M.F.  at  the  terminals,  since  the  motor  is 
supposed  to  be  supplied  from  a  constant  potential  source  having  neg- 
ligible impedance. 

[This  condition  gives,  immediately,  for  the  corresponding  alge- 
braical value  of  &2  the  following 

(25) 


Suppose,  for  example,  a  40  k.w.  motor  connected  to  a  2ooo-volt 
circuit,  its  efficiency  being  77  =  0.85,  and  the  constants  being  R=  2  ohms^ 
and  £=40  ohms.  The  minimum  current  necessary,  at  full  load,  will 
be  about 

40,000 

/=  -     --  =  2  3  .  c  amperes  ; 
0.85X2000 

whence  Z7=  940  volts. 

The  corresponding  E.M.F.  will  be 


£2=  V(2ooo)2+(94°)2~ 2000X47X2=2166  volts. 

The  corresponding  value  of  the  exciting  current  will  be  obtained  by 

reference  to  the  excitation-curve  of  the  alternator. 

V-Curves.     The  foregoing    conclusions,   which    had    already    been 

formulated   by  the  author  in  a  .paper  in  1892  (La  Lumiere  Electrique, 

Vol.  XLV,  pp.  423  and  563),  were  confirmed 
experimentally  by  Mordey  in  1893.  Mor- 
dey's  results  were  represented  graphically 

^  \         /  by  a  curve,  in  rectangular  co-ordinates,  to 

which  the  name  of  V-curve  has  been  given, 
owing  to  its  characteristic  appearance  (Fig. 
gjyg-     30).     This  curve   is   obtained  by  taking  as 
FIG-  30.  abscissae  the  values  of  the  E.M.F.  E2  and  as 

ordinates  the  corresponding   current-values. 

It  is  readily  seen  that  the  current  does  pass  through  a  minimum   Mm 

for  a  certain  value  Om  of  the  voltage. 


DETAILED  STUDY  OF  OFERAT10N  WITH  NORMAL  LOAD  57 

Use  of  Diagram  of  the  First  Kind.  The  diagram  oi  the  first 
kind  lends  itself  more  rigorously  than  the  other  to  the  study  of  volt- 
age and  current  values  when  the  motor  is  working  at  constant  load; 
and  it  also  applies  equally  well  in  the  case  where  the  impedance  of  the 
generator  can  be  neglected. 

It  is  sufficient  to  cause  the  segments  OA2,  OM2,  O" 'A2,  O'"A2,  (Fig. 
31),  representing  the  E.M.F.  E2)  to  vary  in  inverse  ratio  with  respect 
to  the  segments  A2D,  A2D',  A2D",  representing  the  active  current 
Iw,  in  such  manner  that  the  product  E2IW  shall  be  constant  and  equal 
to  the  given  power  P^  If,  from  each  point,  O,  O',  O",  an  arc  of  circle 


is  described,  having  for  its  radius  the  generator  E.M.F.  E\,  their  respect- 
ive intersections  with  the  lines  passing  through  D,  D',  D",  D'"  and 
perpendicular  to  A2Y,  determine  the  load-points  A\,  A\,  A"L,  A'"\\ 
and,  consequently,  the  current-values  /corresponding  to  the  various 
values  of  E2.  This  diagram  also  applies  in  the  case  where  EI  varies 
according  to  some  given  law. 

Predetermination  of  V-Curves.  The  preceding  diagrams  enable 
these  curves  to  be  obtained  by  a  simple  graphical  transformation, 
without  calculation.  It  is  sufficient  to  measure,  on  the  diagram  itself, 
the  radial  vectors  for  corresponding  values  of  E.M.F.  and  current, 
and  to  plot  them  as  rectangular  abscissae  and  ordinates.  Each  of  the 
power-lines  of  Fig.  28,  for  example,  also  gives  a  transformed  curve 


58 


SYNCHRONOUS  MOTORS 


which  is  the  corresponding  V-curve  (Fig.  32).  These  curves  show 
clearly  that  the  V  becomes  more  rounded  as  the  corresponding  power 
P2  increases.  This  fact  is,  moreover,  verified  by  experience,  as  shown, 


£0 


'0 


so 


\'£ 


/- 


FIG.  32. 

for  example,  by  the  curve  in  Fig.  33,  obtained  experimentally  from  a 
synchronous  motor  of  Labour  type  running  at  constant  load  of  5  k.w. 
(half  load). 


^ 

fqooo 

8,000 

$ 

u,  Amperes 

^x 

>  — 

==^ 

^ 

—  -— 

6,000^ 
4,000  ^ 
2,OOO. 

I 

Exciting  Current,  Amperes 

FIG.  33. 

The  sharpest  V  is  obtained  with  no  load.  Theoretically,  the  curve 
^2=0  (Fig.  32)  ought  to  present  an  angular  point  at  the  axis  of  amperes; 
but,  in  practice,  the  power  consumed  by  friction,  eddy  currents 
etc.,  when  running  without  load,  is  not  negligible,  and,  for  these  rea- 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  59 

sons,  the  curve  of  no-load  has  a  form  analogous  to  the  5  k.w.  curve 
of  this  figure. 

It  will  be  noticed  that,  in  practice,  it  is  possible  to  obtain  only  a 
rather  limited  portion  of  curve,  under  load,  because  the  values  become 
more  and  more  unstable  as  they  depart  from  the  minimum  current 
value,  either  way. 

Theoretical  Form  of  V-Curves.  Although,  for  various  reasons 
considered  later,  the  V-curves  always  differ  slightly  in  practice  from 
those  deduced  from  the  diagram,  it  is  interesting  to  determine  their 
theoretical  algebraical  equation. 

If  we  note  that  the  current  7  and  the  power  P2  are  given,  by  eqs. 
(4)  and  (8),  in  terms  of  EI,  E2,  /-,  and  (/>,  it  will  be  seen  that  we  only 
need  to  eliminate  <j>  between  these  two  equations  to  obtain  the  desired 
relation  between  E2  and  /. 

From  eq.   (4),  expanding  and  ^replacing  sin  7-  and  cos  7-  by  their 

•y  r> 

values  —  and  — ,  we  have 


JE12=£22+  (Z7)2  +  2E2[ZI  cos  (/>  cos  r+ZI  sin  0  sin  r] 

(m) 


=  E22+  (Z7)2+  2P2R+  2XVE22I2-P22 
The  equation  obtained  on  squaring  to  remove  the  radical  sign  is 
[(ZI)2+E22-E12+2P2R]2-4X2(E22I2-P22)  =  o. 

This  equation  is  of  the  4th  degree  in  I  and  in  E2.  The  V-curves 
obtained,  in  rectilinear  co-ordinates,  taking  E2  as  abscissae  and  /  as 
ordinates,  are '  therefore  more  or  less  complicated  curves  of  the  4th 
degree.  They  may  be  calculated  by  points  by  solving  with  respect 
to  7,  for  example,  but  the  calculation  is  unnecessarily  complicated; 
and  the  graphical  construction  shown  in  Fig.  31  is  preferable. 
In  the  particular  case  where  P2=o  the  equation  reduces  to 

(ZI)2  +  E22±2XE2I=El2, 

i.e.,  an  assemblage  of  two  easily  constructed  ellipses  whose  axes  inter- 
sect each  other  as  shown  in  Fig.  34.  The  V-curve  for  no  load  is  there- 
fore formed  of  two  branches  of  an  ellipse  which  intersect  each  other  on 
the  axis  of  amperes.  The  sections  of  it  which  are  in  the  useful  portion 
are  almost  rectilinear. 


60 


SYNCHRONOUS  MOTORS 


[Since  there  is  no  term  of  the  first  degree,  these  ellipses  have  the  origin  of  the 
co-ordinates  for  their  center.  If  we  take  as  ordinates  the  product  Z7,  the 
inclination  of  their  axes  is  constant  and  equal  to 

ZI 

—  = tan  45°=  i. 
•  £2 

If,  instead  of  the  product  (Z7),  we  take,  as  ordinates,  the  values  of  7,  the  inclina- 
tion of  their  axis  becomes 


7      Z2- 


FIG.  34. 

This  inclination  increases  inversely  as  the  reactance  X,  i.e.,  a  given  variation 
of  the  exciting  current  produces  a  current-variation^  which  is  all  the  greater  the 
higher  the  inductance  of  the  motor.  This  conclusion  is  supported  by  other  con- 
siderations, as  we  have  already  seen.] 

The  conditions  for  which  the  current  is  a  minimum  are  those  for 
which  the  phase  is  zero  with  respect  to  E\.  They  are  obtained  by  tak- 
ing, on  the  diagram  of  Fig.  27,  all  the  values  situated  on  the  right 
line  AiN.  For  these  ppints  the  triangle  OAiA2  gives 

£22=  E?  +  (Z/)2  -  2EiZI  cos  r 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  61 

In  rectangular  co-ordinates  the  relationship  between  E2  and  7  is 
represented  by  the  hyperbola 1  passing  through  the  point  of  inter- 
section of  the  two  arcs  of  the  ellipse  of  zero-power,  and  having  its  principal 
axis  horizontal  and  situated  above  the  axis  of  abscissae  at  a  height 
equal  to 


The  excitation  which  renders  the  current  a  minimum  must  there- 
fore decrease  at  first  when  the  power  required,  P2,  increases;  and  it 
must  then  increase  more  and  more.  The  greater  the  impedance  the 
sooner  this  increase  will  occur.  These  results  are  confirmed  by  experi- 
ence, but,  in  practice,  this  critical  value  of  E2  may  be  considered  con- 
stant. 

Curve  of  Reactive  Current.  Fig.  32  would  be  incomplete  if  only 
the  total  current  /  were  represented  therein,  because  the  diagram  would 
not  then  give  information  in  regard  to  the  phase  of  the  current.  It 
is  therefore  quite  useful  to  add,  to  the  V-curves,  other  curves  indicating 
the  value  of  the  corresponding  positive  or  negative  reactive  current. 
The  values  of  the  reactive  current  are  deduced  from  the  diagram  by 
constructing  the  lines  corresponding  to  each  value  of  the  phase-angle 
0.  The  curve  of  reactive  current  corresponding  to  P2=2o  k.w.  has 
been  drawn  as  an  example  in  Fig.  32.  In  practice  this  curve  can  be 
plotted  from  measurements  made  by  a  wattmeter. 

Expression  for  Reactive  Current.  The  expressions  for  currents 
which  are  reactive  with  regard  to  EI  or  E2  can  be  calculated  in  like 
manner.  The  reactive  current  as  a  function  of  E2  will  be  obtained 
by  starting  from  eq.  (m).  This  equation  can  be  written 

P9\21 

gj    \+2P2R+2E2XId. 

This  is  also  an  equation  of  the  4th  degree  in  E2,  but  it  can  be  easily 
solved  with  respect  to  Ij,  if  written  in  the  following  form: 


(7  '  p  \ 
--^) 
£-2  I 


1  The  conditions  of  zero  phase-angle  of  the  current  with  respect  to  the  E.M.F. 
E2  would  be  the  same  on  an  ellipse  such  as 


but  they  present  no  special  interest. 


62  SYNCHRONOUS  MOTORS 

For  the  particular  case  P2=o,  this  equation  reduces  to 
Z2Ic2+2XE2Id+E22=E12, 

which  represents  one  of  the  ellipses  of  zero-load  already  determined. 
The  reactive  current  j,  with  respect  to  EI,  is  obtained  by  writing  the 
relation  between  E2  and  EI,  as  it  is  depicted  in  Fig.  26,  thus 

E#=E12+  (ZI)2-2EiZ(i  cos  f+j  sin  ?-), 
or 


With  no  load,  when  P2=o,  this  equation  reduces  to 


The  curve  of  /  as  a  function  of  E2,  drawn  in  rectangular  co-ordinates, 
would  therefore  be  a  hyperbola;    we  can  write 


The  horizontal  axis  is  situated  at  a  distance 


above  the  axis  of  volts. 

These  formulae  enable  the  reactive  currents  to  be  calculated  alge- 
braically with  more  precision,  but  also  much  more  laboriously,  than 
by  the  graphical  method. 

Comparison  of  Outputs  which  the  Same  Alternator  Can  Develop 
with  the  Same  Terminal  Voltage,  when  Used  Either  as  a  Generator 
or  as  a  Motor.  Two  different  cases  have  to  be  considered: 

(i)  Let  us  suppose  the  case  of  an  alternator  connected  to  a  con- 
stant potential  system  of  great  power,  presenting  a  negligible  impedance, 
and  let  the  excitation  be  regulated  in  such  a  manner  as  to  give  the 
minimum  current.  Assuming  a  sufficient  margin  for  regulating  the 
excitation,  the  output  of  the  machine  will  be  limited  in  all  practical 
cases  by  the  value  of  the  armature-current. 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD   63 

Let  r  equal  the  armature-resistance,  x  the  armature-reactance, 
z  the  armature-impedance,  U  the  voltage  at  the  terminals,  7  the  max- 
imum allowable  current.  The  internal  E.M.F.  of  the  corresponding 
generator,  necessary  to  produce  this  current,  will  be  obtained  (Fig. 
35)  by  combining  the  vector  U  with  a  vector  AiA  =  zI  drawn  below 

3C 

U  at  an  angle  =  7-,  such  that  tan  7-=  — .     In  order  that  the  same  machine, 

when  operating  as  a  motor,  at  the  same  voltage,  £7,  may  furnish  the 
maximum  power,  UI,  with  the  same  current,  7,  the  latter  must  have 
the  same  difference  of  phase,  7-,  with  respect  to  U,  but  in  the  opposite 
direction.  The  internal  E.M.F.,  E2=OA2,  will  therefore  be  obtained 
by  compounding  OA  with  a  vector  z7  equal,  but  opposed,  to  the  pre- 
ceding one. 

Fig.  35  thus  shows    that  the  alternator  corresponds  to  the  same 


AS       \ 


FIG.  35. 

electric  power  at  its  terminals  when  operating  as  a  motor  and  as  a 
generator,  and  this  with  substantially  the  same  excitation.  Assuming 
the  losses  to  be  the  same  in  both  cases,  it  is  seen  that  by  increasing 
slightly  the  E.M.F.  E%,  the  same  normal  power  may  be  counted  upon 
for  a  motor  as  for  a  generator.  (The  normal  power  here  designated 
is  the  power  which  the  alternator  can  furnish  on  a  dead  resistance. 
This  is  the  figure  given  in  the  catalogues  of  manufacturers  of  syn- 
chronous motors.) 

(2)  If  now  we  consider  the  case  where  power  is  to  be  transmitted 
to  a  distance,  the  impedance  Z  which  counts  in  the  operation  of 
the  motor  is  that  of  the  entire  circuit  and  it  can  be  notably  superior 


64  SYNCHRONOUS  MOTORS 

to  the  imedance  of  the  armature.     The  sement  ZI  then  extends  as 


far  as  A1  2,  for  example,  making  an  angle  F,  which  is  such  that  tan  jT=  —  ' 

K. 

and  the  E.M.F.,  OA'^  necessary  to  obtain  this  output  may  be  superior 
to  the  preceding  va,lue.  Eq.  (25)  gives  the  value  thereof.  It  is  nec- 
essary to  foresee  this  condition,  in  studying  an  electric  transmission 
project,  whenever  the  impedance  of  the  line  is  of  the  same  order  as 
that  of  the  alternators. 


IV.  INFLUENCE  OF  MOTORS  ON  THE  GENERAL  OPERATION  OF 
AN   A.C.    ELECTRICAL   DISTRIBUTION   SYSTEM 

Effect  of  Current  of  Synchronous  Motors  on  Distribution-Systems. 

There  is  a,  tendency,  at  the  present  time,  and  with  good  reason,  to  use 
synchronous  motors  under  conditions  which  permit  them,  as  if  they  were 
condensers,  to  compensate  for  the  inductance  of  neighboring  motors. 
This  method,  which  consists  in  overexciting  the  synchronous  motors, 
whether  they  be  utilized  for  the  production  of  mechanical  power,  or 
whether  they  be  running  without  load,  appears  to  have  been  first 
advocated  in  1891  by  Swinburn;  and  it  has  been  used  since  by  many 
engineers  in  Europe  and  in  America.  Following  the  advice  of  Dolivo- 
Dobrowolsky  it  was  used  for  the  Bulach-Oerlikon  power- transmission 
in  1893  (see  letter  of  Dolivo-Dobrowolsky  to  the  Elektrotechnische 
Zeitschrift,  4.  Oct.,  1894,  p.  555);  and  Lahmeyer  also  used  it  at  Bocken- 
heim  (see  Deutscher  Verband  der  Elektrotechnischen  Ingenieure,  Con- 
gress of  June,  1894),  where  he  was  thus  enabled,  it  would  seem,  to 
double  the  output  of  the  generators  by  compensating  all  inductance- 
effects  in  the  distributing  system  by  the  reaction  of  synchronous  motors. 
Some  time  after,  Professor  S.  P.  Thompson  called  attention  to  this 
method  ("  Some  Advantages  of  Alternating  Currents  ",  by  Professor  S.  P. 
Thompson,  British  Association,  1894). 

To  study  this  question,  we  must  ascertain  how  the  reactive  current 
varies  with  respect  to  the  voltage  at  the.  terminals  of  the  generator. 

A  polar  diagram  referred  to  the  line-voltage  as  a  fixed  axis  gives 
the  necessary  values  most  accurately.1  Let  us  take  (Fig.  27,  p.  44) 

i  We  use  here  this  form  of  diagram  to  take  the  ohmic  loss  of  the  motor  into 
account.  In  practice,  the  latter  can  be  neglected  without  great  inconvenience, 
•  and  the  diagram  referred  to  the  motor  E.M.F.  E2,  which  is  more  simple,  will  answer 
the  purpose.  We  can  then  see  more  readily  that  the  conditions  corresponding  to 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  65 

on  a  power-line,  P2,  two  output-points,  A2  and  A'2,  situated  respect- 
ively at  the  right  and  at  the  left  of  the  phase-line  AiN;  and  let  us  project 
them  on  that  line,  at  a. 

From  what  has  already  been  seen,  the  lengths  A2a  and  A2'a  will 
represent  the  values  of  the  reactive  currents  corresponding  to  the 
two  outputs;  but  while  the  first,  A2a,  is  lagging  with  respect  to  the 
working  current,  which  is  equivalent  to  the  effect  of  a  self-induction 
with  all  of  its  objections,  the  second,  A2'a,  is  leading,  which  is  equivalent 
to  the  effect  of  a  condenser. 

We  therefore  see  that  for  all  values  whose  representative  point  is 
situated  on  the  left  of  the  phase-line  A\N  there  will  be  a  reactive  leading 
current.  It  is  this  current,  equivalent  to  that  of  a  condenser,  which 
enables  the  lagging  reactive  current  to  be  neutralized,  including  the 
magnetization-current  necessary  for  various  translating  appliances, 
such  as  induction-motors,  transformers,  induction-coils,  etc.,  on  the 
same  portion  of  the  circuit. 

As  we  shall  see  later,  it  is  very  easy  to  calculate  these  compensation- 
phenomena  with  a  view  to  their  practical  application. 

In  studying  the  effects  produced  by  the  reaction  of  synchronous 
motors,  we  should,  at  the  outset,  distinguish  between  two  very  dif- 
ferent things,  i.e.,  the  load  (current)  on  the  line  and  the  load  (current) 
on  the  generating  station  supplying  that  line. 

(i)  Compensation  with  Respect  to  the  Line  or  Circuit.  Many 
ordinary  translating  appliances,  notably  transformers,  arc-lamps,  and 
especially  induction-motors,  have  a  power-factor,  cos  </>,  which  is 
lower  than  unity,  and  which  increases  uselessly  the  heating  of  the  con- 
ductors of  the  distributing  system  cr  of  the  line,  in  the  case  of  a 
simple  transmission-system.  [The  loss  of  voltage  in  the  line  is  likewise 
increased  in  greater  proportion  than  the  current,  in  consequence  of  lag,  as 
the  author  has  shown  elsewhere.  See  articles  on  "  Line-inductance  for 
Alternating  Currents  ",  VEdairage  Electrique.  Oct.,  Dec.,  1894.] 

The  aim  of  the  compensation  sought  should  be  to  increase  this 
power-factor  as  much  as  possible,  by  producing  the  reactive  currents 
locally  instead  of  transporting  them  over  the  line. 

The  evaluation  of  these  currents  is  not  difficult,  owing  to  the  fact 
that  in  the  translating  appliances  just  mentioned  the  magnetizing 

points  situated  on  the  left  of  the  reference-line  OY  are  those  which  produce  a 
leading  current  with  respect  to  the  voltage  at  the  terminals.  It  is  therefore  on 
the  left  of  OF  that  the  motor  plays  the  role  of  a  condenser. 


66  SYNCHRONOUS  MOTORS 

(or  reactive)  currents  are  practically  constant  at  all  loads  and  can  even 
be  predetermined,  at  least  roughly,  according  to  the  output  of  the 
appliances,  without  knowing  exactly  what  these  are.  Let  Jd  be  the 
total  value  of  the  reactive  currents,  and  Jw  be  the  total  corresponding 
active  current,  at  any  given  instant. 

.  To  neutralize  completely  the  lag  of  the  line,  it  would  be  necessary 
to  produce,  by  means  of  the  synchronous  motors  in  that  part  of  the 
line,  a  reactive  current  which  is  leading  in  phase  and  equal  to 


(in  absolute  units).  But  it  is  not  usually  necessary  to  fulfill  strictly 
this  condition,  because  it  may  be,  as  a  rule,  considered  sufficient  to 
have  a  power-factor  0.95  to  0.97,  corresponding  to  reactance-factors 
(tan  -f)  ranging  in  value  from  1  to  J,  respectively.  Let  us  take, 
for  example,  tan  7'=^-.  It  is  then  sufficient  to  satisfy  the  equation 


Jw+i     3' 

in  which   i   equals   the  active   current   absorbed   by   the  synchronous 

motors,  and  which  must  be  added  to  that  of  the  apparatus,  Jw  (useful 

load). 

From  this  we  have 


This  means  that  the  reactive  current  which  the  synchronous  motors  must 
produce  to  render  the  reactance-effects  of  neighboring  motors  practically 
negligible  need  not  be  equal  to  the  current  actually  consumed  by  these 
appliances,  but  may  be  reduced  more  and  more  as  their  useful  load  Jw 
is  increased.  Owing  to  this  circumstance,  one  or  more  relatively  small 
synchronous  motors  will  suffice,  in  practice,  to  neutralize  the  harmful 
effects  of  reactance  on  the  line. 

Let  us  suppose,  for  simplicity,  that  there  is  only  one  such  compensa- 
tor-motor, and  that  it  is  also  serving,  at  the  same  time,  as  a  motor 
doing  mechanical  work.  Its  total  armature-current  will  be  equal  to 
the  resultant 


•o    t     *o 
iZ  JL  *Z  — 


t*+r= 


The  current  I  being  limited  by  the  heating  of  the  motor,  the  useful 
current  i  and,  consequently,  the  useful  corresponding  power  P2,  will 


DETAILED   STUDY  OF  OPERATION  WITH  NORMAL  LOAD  67 

have  to  be  all  the  lower  the  greater  the  value  of  /.     If  it  is  satisfactory 
for  example,  to  take  j=$i,  the  increase  of  /  will  be  almost  negligible. 
The  graphical  determination  of  /  and  E2)  is  obtained  readily  from 
the  fundamental   diagram  (Fig. 
26)   by    constructing    (Fig.    36), 
according  to   the  scale  of  volts, 
the  lines  Zi  in  the  direction  AiN, 
and  Zj  in  the  perpendicular  di- 
rection, and   by  drawing  A  \A% 
and  OA  2.     If  7  and  i  are  known 

A  X  C     \  tX 

beforehand,    E2  and    i    can  be  FlG  ^6 

deduced   inversely.     [All  that  is 

necessary  is  to  project  the   broken  line  A2A^O  on  the  line  A\N  and 

on  the  line  perpendicular  thereto;  the  two  projections  are 

Zi—Ei  cos?- 
and 


The  sum  of  their  squares  gives  E22,  from  which  we  deduce  the  formula, 
(26)]. 

This  diagram  also  gives  the  algebraical  expression  of  E2.     thus 


(26) 


From  this,  given    the    excitation-curve  of  the    motor,    the    necessary 
ampere-turns  can  be  deduced. 

In  order  that  the  compensation  may  be  really  advantageous  for 
the  line,  it  is  necessary  that  the  resultant  currents  should  be  notably 
weaker  than  before,  i.e.,  we  must  have 


V(Jw  +  if  +  (Jd  - 

In  practice,  compensation  of  this  kind  is  especially  advisable  when 
synchronous  apparatus  (motors  or  converters)  is  already  connected  on 
the  distribution- system,  and  serves  for  other  purposes;  because,  all  that 
is  then  necessary  is  to  increase  slightly  the  excitation-current  of  each  of 
them,  which  involves  no  special  expenditure  of  energy  other  than  a 
slight  supplementary  loss  in  the  fields  and  in  the  armature.  The 
companies  which  operate  alternating-current  systems  should  therefore 


68  SYNCHRONOUS  MOTORS 

seek  to  encourage  as  much  as  possible  the  use  of  such  apparatus  by 
their  patrons. 

When  no  such  apparatus  exists  on  the  system,  it  may  be  desirable 
to  place  synchronous  motors  running  without  load  at  the  distributing 
centers  for  the  special  purpose  of  furnishing  reactive  current.  Let 
us  see  what  the  nominal  power  of  such  a  motor  should  be. 

The  reactive  current  i  being  then  only  the  current  necessary  to 
keep  the  apparatus  in  motion,  if  we  designate  by  p  the  sum  of  the  losses 
without  load  (i.e.,  friction,  hysteresis,  eddy  currents  and  power 
necessary  for  excitation),  we  will  have,  approximately, 

._p+rl2 
~ET 

If  the  apparatus  is  efficient  i2  will  be  very  small  with  respect  to 
j2  and  consequently  we  can  write,  as  an  approximation, 


[In  a  good  motor  the  active  current  with  no  load,  i,  does  not  exceed 
o.io  or  0.20  of  the  full-load  current.  The  lag  of  the  current  of  an  over- 
excited motor  running  without  load  can  therefore  approach  very  near 

—  ,  without  nevertheless  approaching  as  closely  thereto  as  would  the 

current  of  a  condenser.] 

Since  0  is  very  small,  under  these  conditions,  E2  becomes  approx- 
imately 


The  power  for  which  the  motor  is  rated  by  the  manufacturer  is 
equal  to  the  product  of  the  ordinary  E.M.F.  (usually  with  a  margin  of 
15  to  25  per  cent  increase),  by  the  normal  current.  For  example,  under 
this  hypothesis,  it  would  be  possible  to  use  a  motor  having  a  catalogue 

7™1        /  Z7       \ 

voltage  of  -^-  (  more  generally,  —  ~  j  and  having  the  normal  current  7. 
Its  rated  power  will  be 


in  which  j  has  the  value  already  given.  This  value  of  j  shows  that,  in 
an  installation  comprising  a  certain  number  of  machines  in  permanent 
operation,  the  capacity  of  synchronous  motor  necessary  for  compensa- 
tion may  be  taken  lower  in  proportion  as  the  load-factor  will  be  higher, 


DETAILED   STUDY  OF  OPERATION  WITH  NORMAL  LOAD  69 

and  that,  if  sufficient  compensating  effect  has  been  obtained  at  light 
loads,  the  compensating  effect  will  be  still  better  with  heavier  loads. 
First  Numerical  Example.  Assume  a  distribution- system  supply- 
ing 500  k.w.  of  transformers  at  5000  volts.  The  magnetizing  current 
is  a  small  fraction  of  the  full-load,  current,  varying  usually  from  2  per 
cent  to  5  per  cent,  according  to  whether  the  transformers  are  large  or 
small.  Let  the  average  be  taken  at  3.5  per  cent.  The  full-load  current 
being  100  amperes,  the  total  magnetizing  current,  which  is  approximately 
constant  at  all  hours  of  the  day,  will  be  3.5  amperes.  The  power- 
factor  with  no  load  being,  on  the  average,  0.70,  the  active  current  with 

no  load  will  also  be  3.5,  and  the  resultant  current  will  be  —  =  5  amperes. 

To  neutralize  the  lag  by  means  of  a  synchronous  motor  having  a  rated 
efficiency  of  0.80  when  running  without  load,  it  would  be  necessary 
to  produce  a  reactive  current  of  3.5  amperes  at  the  expense  of  0.7  ampere 
of  active  current  corresponding  to  the  resistance-losses.  The  current 
taken  from  the  distribution-system  would  then  be  3.5  +  0.7  =  4.2  amperes, 
instead  of  5  amperes.  This  saving  is  not  sufficient  to  justify  the  pur- 
chase and  maintenance  of  a  motor  intended  to  serve  solely  as  a  com- 
pensator; but  the  use  of  a  condenser  would  not  be  much  more  advan- 
tageous, as  the  current  would  always  remain  slightly  above  3.5  in 
consequence  of  dielectric  hysteresis. 

Second  Numerical  Example.  Assume  a  distribution-system  com- 
prising, at  the  end  of  the  line,  where  the  tension  is  2000  volts,  some 
translating  devices  composed  of  200  k.w.  of  induction-motors  and  100 
k.w.  of  synchronous  motors,  the  latter  having  a  total  reactance  of  15 
ohms  and  a  resistance  of  0.8  ohm.  Assume  the  synchronous  motors 
to  have  an  efficiency  of  80  per  cent,  the  induction -motors  to  have  a 
magnetizing  current  equal  to  one-third  of  the  normal  current,  an 
efficiency  of  85  per  cent,  and  a  power-factor  of  0.95,  at  full  load. 

[The  magnetizing  current  generally  varies  between  one-third  and 
one-quarter  of  the  load-current  in  polyphase  motors  and  between  one- 
half  and  one-third  in  single-phase  motors.] 

The  problem  assumed  is:  to  compensate  the  effects  of  the  lagging 
current  in  the  induction-motors  when  running  with  no  load,  and  also 
with  a  quarter-load,  it  being  assumed  that  all  the  motors  are  constantly 
in  operation. 

The  total  current  necessary  for  the  induction-motors  at  full  load  is 

200OOO 

124  amperes. 


0.85X0.95X2000 


70  SYNCHRONOUS  MOTORS 

The  reactive  current  will  be,  at  full  load,  approximately 

124 
= =42  amperes. 

It  can  be  assumed  that  the  corresponding  active  current,  at  no  load, 
is  10  per  cent  of  the  total,  or  12  amperes. 

Since  the  synchronous  motors,  in  this  case,  run  without  load,  it 
would  be  necessary,  in  order  to  compensate  absolutely  for  the  phase- 
lag,  to  make  them  produce  also  a  reactive  current 

j=42  amperes; 
and  since  their  active  current  attains,  at  most, 


.0.20X100000 

— =  10  amperes, 
2000 


these  motors  will  take  a  maximum  current  equal  to 


7=  V(io)2+(42)2=42.i  amperes, 
while  their  normal-load  current  might  attain 


100000 

=  62.5  amperes. 


0.80X2000 

The  E.M.F.  is  given  by  the  approximate  formula 
£2  =2000 +(15X42)  =  2630  volts. 
The  line-current,  which,   without  compensation,   would  be 


V/(io+i2)2+ (42)2=  50.5  amperes 

(or  at  least  42.1  amperes,  if  the  synchronous  motors  were  not  operating) 
would  thus  be  reduced,  by  the  compensation,  to 

10+12=22  amperes. 

It  might  even  be  reduced  to  19  amperes,  if  only  two-thirds  of  the 
synchronous  motors  were  being  operated. 


DETAILED   STUDY  OF  OPERATION  WITH  NORMAL  LOAD  71 

Let  us  suppose,  now,  that  each  group  of  motors  shall  furnish  50 
k.w.  The  active  current  for  the  induction-motors  can  be  estimated 
.at  25  amperes,  plus  8  amperes  for  losses,  or  in  all  33  amperes;  and 
the  active  current  for  the  synchronous  motors  can  be  estimated  at 
25  amperes,  plus  the  losses,  which  are  at  most  10  amperes,  or,  in  all, 
35  amperes. 

The  total  active  current  is  therefore  33  +  35=68  amperes,  while 
the  reactive  current  of  the  induction-motors  remains,  as  before,  42 
amperes.  The  resultant  current  is 


v  (42)2+ (68)2=92. 5  amperes, 

and  it  could  not  be  less  than  68  amperes. 

Suppose   it  be  required  to  bring  up  the  power-factor  to  0.95,  i.e., 
to  bring  the  current  down  to   72  amperes.     It  will  be  sufficient  to 

AQ 

reduce  the  reactive  current  to  —  =23  amperes  by  producing  a  current 

o 
out  of  phase  which  is  equal  to 

7=42  —  23=  19  amperes. 

The  current  of  the   synchronous  motors  will  then  become  equal 
to  about 


V  (19)2 +(3 5)2=  40  amperes 

instead  of  35    (i.e.,   it  would  be  slightly  increased).     We  will  have 
tan  7-=  Jf  =0.543  and  the  E.M.F.  will  be  equal  to 

£2=2000^  i +  ^^"+^^(19X0.4771 -35X0.8788) 
=  1728  volts. 

instead  of  1486  volts,  which  the  motor  should  develop,  when  the  phase- 
angle  f  equals  zero. 

It  is  seen  that,  by  overexciting  to  a  maximum  of  30  per  cent,  it  is 
possible  to  compensate  for  the  full-load  reactions,  without  overloading 
the  synchronous  motors.  To  obtain  the  same  result  by  means  of  a 
motor  running  without  load  it  would  be  necessary  to  use  an  apparatus 
having  a  rating  of  about  40  k.w.;  which  would  not  be  very  practical. 


72  SYNCHRONOUS  MOTORS 

Economic  Study  of  Compensation  for  the  Line  by  Means  of 
Motors  Running  without  Load.  For  simplicity,  let  us  suppose  the 
following  to  be  constant:  the  voltage  at  the  sending  and  receiving 
ends,  the  power  to  be  transmitted,  and  the  loss  of  energy  in  the  line. 
It  is  required  to  determine  the  economy  resulting  from  compensation 
by  overexcited  synchronous  motors  running  without  load,  for  the 
different  values  of  cos  </). 

Two  points  have  to  be  considered: 

(1)  The  saving  in  the  capital  invested; 

(2)  The  saving  in  annual  cost  of  operation. 

These  two  points  lead,  as  will  be  seen,  to  conclusions  which  are 
appreciably  different: 

(i)  Saving  in  Cost  of  Equipment.  Let  (for  a  single-phase  current), 
jEo^the  voltage  at  the  transmitting  end  of  the  line;  £i  =  the  voltage  at 
the  receiving  end;  £2=  the  E.M.F.  of  the  synchronous  motors;  Iw= 
the  useful  active  current;  i  =  the  active  current  of  the  synchro- 
nous motors;  I,j=Iw  tan  <j)=  the  reactive  current  when  there  is  no 
compensation;  /=the  reactive  current  of  the  synchronous  motors; 
/i=the  resultant  current  of  the  synchronous  motors;  ^0=the  cost 
of  the  generators  per  kilo  volt-ampere  ;  pi=  the  cost  of  the  synchronous 
motors  per  kilovolt-ampere  ;  p2=  the  price  of  line-copper  per  kilo  volt- 
ampere  transmitted. 

The  price  of  line-copper,  when  there  is  no  compensation,  is  given 
by  the  formula 


.  ........ 

(which   includes    the  return-conductor).     In   this  formula  the   factor 
p2  is  equal  to 

.....     (28) 


wherein  /=  length,   in   kilometers;    b=  price   of   copper  per   kilogram, 

(assumed  =2. 50  francs). 

s=the  percentage  of  loss  allowed  in  the  line;    £0=the  initial 
E.M.F.  in  volts. 

The  voltage  being  supposed  constant  at  the  transmitting  and  receiv- 
ing ends,  it  follows  that,  in  order  to  transmit  the  same  power  with  the 


DETAILED   STUDY  OF  OPERATION  WITH  NORMAL  LOAD  73 

same  losses  in  the  line  for  the  different  values  of  cos  ^>,  the  weight  of 
the  line-copper  will  have  to  vary. 

The    cost,  in  francs,1  of  the  line-copper,  when  there  is  no  compen- 
sation, is 


and,  when  there  is  compensation-  it  is 


cos  <> 
The  cost  of  the  generators,  without  compensation,  is 


and,  with  compensation, 


COS  (f) 

Finally,  the  cost  of  the  synchronous  motors  is 

#1^1/1, 

wherein 


The  total  saving  in  cost  of  equipment,  as  the  result  of  compensa 
tion,  will  be  (in  francs)1 


(29) 


1  The  corresponding  cost  value  in  dollars  will  be  given  by  the  same  formula 
by  introducing  the  factor  0.20  in  the  formula.  The  same  remark  applies  to  all 
the  formulae  which  follow  in  which  cost-values  are  given  in  francs. 


74  SYNCHRONOUS  MOTORS 

(2)  Saving  in  Annual  Operating  Cost.  The  annual  operating 
expenses  include  depreciation  and  interest  on  the  capital  invested, 
and  the  cost  of  the  extra  energy  expended  in  the  generators  and  syn- 
chronous motors. 

The  energy  expended  in  the  line  is  supposed  to  be  the  same  with 
or  without  compensation. 

Let  us  assume  the  following  figures: 

Depreciation  of  copper 3    per  cent; 

of  machines 10  per  cent; 

Interest  on  capital 5    per  cent; 

Price  of  one  k.w.-hour  of  energy 10  centimes  (2  cents). 

Let  I'  equal  the  resultant  current  at  the  generators  when  there  is 
compensation;  and  let  us  assume  that  the  resistance-loss  in  the  gen- 
erators, at  full  load,  amounts  to  4  per  cent  of  the  normal  effective 
output. 

Under  these  conditions,  the  saving  in  resistance-loss  in  the  generators 
when  there  is  compensation,  will  be  (in  francs) 


wherein  h  is  the  total  number  of  hours  of  operation  of  the  installation. 
The  saving  in  interest  and  depreciation,   for  the   machines,   will 
be  given  by  the  sum 


(31) 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  75 

Finally,  the  cost  of  the  energy  used  by  the  synchronous  motors  will  be 
(in  francs) 


Numerical  Example.  Let  us  apply  the  preceding  formulas  to  a 
particular  case. 

Assume  a  transmission  of  500  k.w.  to  a  distance  of  24.75  kilo- 
meters with  5000  volts  at  the  receiving  end,  and  a  loss  of  10  per  cent  in 
the  line. 

Suppose  it  be  desired  to  effect  complete  compensation  by  means 
of  synchronous  motors  running  without  load.  These  motors  must 
produce  a  reactive  leading  current 

/=  Iw  tan  (f>. 

Let  us  suppose,  moreover,  that  the  active  current  of  the  motors,  when 
running  without  load,  is 

i  =0.10/1. 
Since  we  have 


we  find 


Iw  tan 

1= := 


iw  ran  9 

and  /i  =  — . 

Vo.99 

Formula  (28)  gives 

/>2=  436.36  francs  ($87.27). 

The  voltage  at  the  receiving  end  being  5000  volts,  and  the  loss  in 
the  line  10  per  cent,  the  initial  voltage  must  be 

£0=-l=555o  volts. 

Let  us  assume  the  cost  x>f  the  generators  and  motors  to  be 
pQ=pl=IOo  francs  ($20)  per  kilo  volt-ampere. 


76 


SYNCHRONOUS  MOTORS 


O* 

H 


w    o 

PQ      H 

H     O 
U 


rH         K«|       Cti 


^  a 

3  "c 

O    'oj 


I.  2 

M(D 
t  i 


||||.3 

<i  ti  d§.". 


u, 


oOO 

cs    vO 


1        1        1        1        1      +    + 


O    NO 

&  ^ 

co    to 

+    + 


O     10    ro    O 

t^    r<5    M    OO 
to    O      ^D    ON 


OOOOOioOO 


O     O 

o    o5 


O     to    O     O 


to    ro    •* 
•*     ON    ^t- 

ON    ro    H 


o    o 

O    00 

ON   00 


1        1        1        1 


OsOO     "t 
O      QsOs 


Tt     ON  CM 

M     00  ON 

M        CO  tO 

<%     c?  c? 


++++++ 


O  to 

O  to 

to  to 

to  to 

NO  O 


0      0     0      0     O 


+ 


Tj-toroONto^N     10 

ooc?o:>oo>"oo" 


H     t-ON'st-c 

<N      rO^iOvOOO     O>.    M 

to    IO'  to    to     IO     to    to  ^O 


t^    CM     O     O     O 

NO"'  NO"    £-    f^  ^- 


t^-  OO     O     H     ro    to 


10    10     O     VO      H 

CM     t^    ^    CM'     -3- 


ooooooo 


OO      ^>*    i>*  NO 

o    o    o    o 


0     O     0      O 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  77 


TABLE  II 
SAVING  IN  OPERATING  COST 

Saving  in  Cost  of  Operation  per  Year. 

Assuming  2000  Working  Hours 
per  Year. 

,_„,„_ 

" 

0  «-«-0  M   0  0  0  tor-Ot       rr^ 

l?ll«SI§!H?§iail 

1  1  1  1  1  1  I  1  1  1  1  1  +  +  + 

Annual  Loss  of 
Energy. 

•sao^oj^  snouojipUA'g  uj 

* 

oo^oooooooooooo 

S«o£<8«  «££&&8228 

1  1  1  1  1  1  1  1  1  1  1  1  1  1  I 

•SJO}BJ9U9Q 

-J 

*«o«*^p?«o*oooo^ 

Nrrv2«^ns^j;^^^a^o; 

i++++++++++++;; 

Assuming  3000  Working  Hours  per  Year. 

jBrtuuy  jo  JB^OX  ^UB^nsg^j 

« 

^J«^g«J«l^gg 

1   1   !    1    1    1    1    1    1   1   1   1   1   1  + 

Annual  Loss  of 
Energy. 

,        0001 

- 

»    '  o      'o'^du^io-oo'diod 

OI  *O 

:sao;oj^  snouoatfouAg  uj 

H   t*  0  000    H    CO  fO  <s    m<3    H  00   M   10 

1      1      1      1      1      1      1      1      1      1      1      1      1      1      1 

1    li~  (  1  1  •  — 

^0^.0^000.0,00 

LA///      A     x     /J^/g-fo-o 

l+++++++++++++l 

Interest  and  Deprecia- 
tion. 

•So'o-f-oi'o  :  sguitjoB  j^  jo^ 

* 

^.s^^^^s?^?^? 

1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 

•So'o+fo'o  :j9ddoj)jo^j 

^ 

O\                        >000    t^.  Tf          Tf                               <S 

rfOOO   M   OO   Ot^-iooOOO  MOO  t^oO 
M   cs  %0   «  00  r-.  t-vO^  r-  0  0  ^^^  Oj 

-,/=*+«/ 

:  (uom2SU9duio;[)  1^/32)  ^ugjjn^  3uirj 

* 

r^ionM         lOl-t-ro 

Tru?r?cru?rr^u?o(?c!t:Trf!rr° 

'T7Z  +  *H  =  Z3 
isjo^oj^  snouoatpuAg  jo  '^'J^'3  Suipuodsojjo^ 

\n  ir>  in  irj  to  in  vo\O  'O  \O  ^O  t^*  t^-  r*oO 

'Ijl  '0  =  1, 

:sjoioj^  snouo-iupuAg  jo  ^ugjanQ  9A]ioy 

Q 

ro^^H         lo^r-ro 

66-0^ 

'0 

f^W    OOOO    M    N    COO    lO^OOO    H    N 

I      ~*2 
:sjo}oj^  snouojijouAg  jo  ^ugjjnQ  iuBijnsg^j 

:  p9ZT['BJ}n9JiJ  9C[  O^  ^U9JJUQ  9Al}0'B3"£[ 

GQ 

J?  0°°    10  Jo00"0    °*  ?                      00    N 

c.roco^^^OO^rO^^cj 

•<f>  SOO  'JO;OBJ-J9MO(J 

- 

O\  ON  ON  O\  O\00  00  00  t-  t^\O  O  »o  »o  TJ- 

ooooooooooooooo 

78 


SYNCHRONOUS  MOTORS 


l-i  w 

W  i-I 

CQ  U 

H  g 


•3UTABS 


snouojiptiAg  jo 
oq 
ssoq 


•SJ01BJ9U9Q  JO 

" 


•S3UIt[OBJ\[ 

joj  uoi^Bioajd 

-9Q  pUB  1S9J91UI 


joj 

-9Q    pUB 


snouoatpuA'c; 


•19 

snouojipuAg 
jo  •  j'  W3  3uipuods9jao3 


snouoaqouAc 


<t>  SOO 
•  ~ 


^uo 


snouojtpuAc; 


•f  503 


OOOOOOt^OO 


1    1    1  +  +  +  +  +  + 


1    1    1    1    1    1   1   1 


oooovooooo\ooo 


t^fO^^O     O     fO 

M       CO   \O       O     OO       M 


O     O     OO     ro-^-0000 
O\    ON    O     10    H     f^OO     H 


vo     M     TJ-  00 


QioO     O     O  O     Q     10    10 

10  OO     10    O^  vO  10    O^    ^    vO 

H        t^O\VOOO  ^M        MOO 

M      W  rj-  O    OO     O 

i  i  I  i  i  i  i  I  T 


OuoOOOOmOO 


rofOO     t^M     t^    Tt    <M     w 


H       W       CO 
1  1  1  1  1  1  1  1 


O     10  O     J>-00 


OiorOOOOOOOO 
O     O     O     O     O     O     O     O     O 


DETAILED   STUDY  OF  OPERATION  WITH  NORMAL  LOAD  79 

To  estimate  the  size  of  the  motors  required,  we  will  suppose  them 
constructed  for  an  E.M.F.  equal  to  that  which  they  should  produce 
when  run  as  compensating  machines. 

For  the  sake  of  economy,  the  manufacturers  tend  more  and  more 
to  increase  the  magnetic  density,  i.e.,  to  '  saturate  "  their  machines, 
and  it  is  scarcely  prudent,  at  present  to  count  upon  being  able  to  increase 
very  much  their  counter  E.M.F.  by  increasing  the  excitation. 


3SOjOOC 
300.00 
250.000 
200.000 
150.000 

/ 

/ 

60,000 

/ 

b 

/ 

? 

30,000 
40,000 
SO.OOff 
20.000 
10,000 

CO 

P 

0,000 

20,000 
3.0.000 
40.000 

I 

1 

$ 

¥ 

•  X 

/ 

V 

$ 

0, 

> 

\v 

rQ 

y 

•  C1 

^ 

7 

f  A 

y 

50.000 

1° 

2 

\ 

/ 

fft 

^ 

<oi 

<y 

(til 

$s 

/ 

s 

/ 

+  2 

^ 

J^» 

3^ 

/ 

^ 

"x^ 

^ 

£* 

r    c. 

—^ 

__£/£ 

"80 
pov 

tr 

i£j 
1 

05  (t 

"60 

<?(• 

050 

or 

H4 

^ 

^s> 

o* 

^ 

^ 

?£" 

100000 

s 

^ 

Q&* 

S{ 

s^ 

1  50  000 

J^f 
\° 

* 

^ 

200.000 

\ 

FIG.  37. 


From  the  calculations  corresponding  to  the  values  of  cos  (j)  com- 
prised between  i  and  0.40,  we  obtain  the  curves  given  in  Figs.  37  and 
38.  The  curves  in  Fig.  37  give  the  saving  in  cost  of  equipment  and  the 
curves  in  Fig.  38  give  the  saving  in  annual  expense  of  operation.  It 
will  be  seen  that  if,  from  the  standpoint  of  cost  of  equipment,  the  use 
of  motors  running  without  load  becomes  rapidly  advantageous  as 
cos  (f>  decreases,  it  is  different,  in  any  particular  case,  from  the 
standpoint  of  the  annual  cost  of  operation. 


80 


SYNCHRONOUS  MOTORS 


The  depreciation  of  the  machinery,  which  is  higher  than  that  of 
copper  and  also  the  consumption  of  energy  required  by  the  motors, 
(which  latter  is  not  compensated  by  the  decrease  in  the  losses  of  the 
generators),  have  the  effect,  together,  of  offsetting  the  saving  made 
in  copper,  even  through  it  may  be  somewhat  important. 

Economy  of  Compensation  for  the  Distributing  System  by  Means 
of  Synchronous  Motors  Running  with  Load.  There  may  be  some 


30.000 

£5000 
20000 
15000 
10000 
5000 

«n       o 
o 
c 

O   5000 
U. 


10.00 


16000 


20000 


25000 


30000 


^S 


ftt 


6000 


5.000 


t.ooo 


3.000 
2,000 
1.000 


o 

LOOcP 

2,000 
3.000 
4.000 
5.000 
6.000 


FIG.  38. 


hesitation  in  regard  to  the  choice,  in  this  case,  because  the  efficiency 
of  synchronous  motors  is  generally  lower  than  the  efficiency  of  induc- 
tion motors,  and,  besides,  in  order  to  produce  a  leading  current,  they 
must  have  a  counter  E.M.F.  higher  than  that  of  induction-motors. 
The  effect  of  this  is  to  increase  their  rating  and  their  cost.  It  is  there- 
fore better,  in  each  particular  case,  to  consider  whether  the  use  of  syn- 
chronous motors  is  advantageous  or  not,  from  an  economic  point  of 
view,  due  consideration  being  given  to  the  necessities  of  the  case. 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  81 

Suppose  a  long-distance  power-transmission  project  in  which  it 
is  possible  to  connect,  with  the  distributing  system,  either  some  induc- 
tion motors  working  with  a  certain  lag,  or  else  an  equipment  composed 
of  one-half  induction  motors  and  one-half  synchronous  motors. 

The  line-current,  with  the  induction  motors  alone,  and  with  a  phase- 

angle  <j),  would  be  equal  to  —  —7.     With  half  the  capacity  in  synchro- 

nous motors  added,  and  effecting  complete  compensation,  the  current 
would  be  equal  to  Iw. 

It  should  be  noted  immediately  that  the  loss  by  resistance-heating 
in  the  motors  will  be  exactly  the  same  in  the  two  cases,  because  the 
magnetizing  current  of  the  synchronous  motors  has  the  same  value 
as  the  reactive  current  of  the  induction-motors,  it  being,  however,  a 
current  leading  in  phase. 

Under  these  conditions,  in  order  to  maintain  the  loss  of  energy 
constant  in  the  line,  for  different  values  of  cos  <p,  the  cost  of  line- 
copper,  when  the  induction-motors  alone  are  used,  should  be 


and,   with  complete  compensation  by  means  of  synchronous  motors 
which  carry  half  the  load,  the  price  of  line-copper  would  be 


For  equal  losses  the  cost  of  line-copper  is  proportional  to  the  square 
of  the  current;  the  saving  effected  in  the  cost  of  copper,  by  introducing 
synchronous  motors,  would  therefore  be 


iw        2w        w 
p2cos</>       cos  (ft        Iw2   ' 


CO 


The  cost  of  generators,  without  compensation,  would  be 


cos 


82  SYNCHRONOUS  MOTORS 

and,  with  compensation,  it  would  be 


The  saving  in  cost  of  generators,  in  the  second  case,  would,  there- 
fore, be 

/     i 

\COS  (f> 

Let  e  equal  the  counter  E.M.F.  of  synchronous  motors  operating  with 
a  magnetizing  current  —  tan  (f>  which  is  lagging.     Let  e^  equal  their 

counter  E.M.F.  with  the  same  magnetizing  current  leading  in  phase. 
The  values  of  e  and  e\  are  deduced  from  formula  (26)   which  gives 


sn   ~  cos 


in  which  E\  is  the  potential  difference  applied  at  the  terminals  of  the 
motor  and  I  the  resultant  current. 
•    The  nominal  power  of  the  motors  and,  consequently,  their  cost, 

would  be  increased,  in  the  second  case,  in  the  ratio  —  ,  and  the  increase 

e 

in  purchase-price  of  the  synchronous  motors,  resulting  from  the  intro- 
duction of  compensation,  and  on  the  assumption  that  the  price  per 
k.w.  of  the  synchronous  motors  and  induction-motors  is  equal,  would 
be 


.    (          Iw  Iw      \ 

Pi  (  ei  -  T  ~  e  -  r  ]  • 
\    2  cos  9       2  cos  9/ 


Let  us  see,  now,  what  would  be  the  saving  in  annual  operating  expenses: 
The  saving  in  interest  and  depreciation,  for  copper,  is 


The  saving  in  interest  and  depreciation,  for  the  machines,  is 

[p^E^Iw[  -  -7  —  i  )  —pi[ei  --  ^—r—e  -  ^—  r  )    (0.10+0.05). 
-  \COS<£         /  \       2COS0  2COS0/J 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  83 

The  saving  in  loss  by  resistance-heating  in  the  generators  (which 
is  proportional  to  the  square  of  the  current),  is 


o. 


Finally,  we  will  assume  the  efficiency  of  the  synchronous  motors 
to  be  8  per  cent  lower  than  that  of  induction-motors,  which  is  rather 


MWMW 

450000 
400000 
550.000 
300000 

0,250.000 
u 
c 
^200.000 
u. 
150000 

100.000 
50.000 

/ 

90,000 
80,000 
70,000 
60.000 
50,000  ri 

1 

40,000^ 
30,000 
20,000 
10,000 
0 
1000 

L 

> 

7 

7 

7 

r 

V 

y 

^ 

* 

'it 

/ 

<< 

f 

/^ 

oV 

4 

tf 

' 

// 

0 

7 

x^ 

\ 

//( 

y 

^V 

J 

A 

,.<>> 

y 

\ 

// 

?x 

^ 

*  pf 

er^ 

x^ 

—  ^ 

^ 

rr 

sti 

Ll— 

rp 

0 

~ 

~ch 

=, 
/or, 

= 

l/7/t 

y  ' 

= 

f^/7<: 

A^ 

. 

5 

^onnn 

Me 

/£?/ 

5 

FIG.  39. 

exaggerated,  because  synchronous  motors  are  actually  constructed 
whose  efficiency  approaches  that  of  induction-motors.  We  will 
assume,  however,  that  the  efficiency  is  lower,  in  order  to  justify  the 
assumption  of  equality  of  cost  per  k.w.  for  the  two  classes  of  motors. 
The  increase  in  annual  cost  of  the  energy  expended  in  the  synchro- 
nous motors,  resulting  from  that  difference  in  the  efficiency,  would  be 


jX— XA, 

2  1000 


84 


SYNCHRONOUS  MOTORS 


in  which,  as  before,  h  is  the  total  number  of  hours  that  the  plant  operates 
per  annum. 

Numerical  Example.  Let  us  take  the  same  case  of  power-trans- 
mission already  mentioned  in  which  the  total  power  to  be  transmitted 
k.w.  is  500  the  distance  10  kilometers,  and  the  potential  difference, 
at  the  receiving  point,  5000  volts. 

Let  us  assume  that  the  total  impedance  of  the  synchronous  motors 


50.000 
45000 
40500 
35,000 
30,000 
25.000 
80.000 


16.000 


10.000 


5000 


5000 
6.000 


£ 


10,000 
9.000 
8,000 
7.000 
6,000 

5,000  ^ 
M 

4,000  rg 
Q 

5,000 
2,000 
1,000 

140 


Annual  loss  «f  energy  dice  to  lev  efficiency 
of  synchronous  motors. 

FIG.  40. 
TABLE  III 

is  15  ohms,  and  that  the  plant  is  in  operation  a  total  of  3000  hours 
per  annum.  On  substituting  the  corresponding  numerical  values  in 
the  preceding  formulae,  and  making  the  calculations  we  obtain  data 
from  which  Table  III  (page  78)  has  been  prepared,  the  results  being 
shown  graphically  in  Figs.  39  and  40. 

The  line  parallel  to  the  axis,  in  Fig.  40,  represents  the  annual 
loss  of  energy  due  to  the  decrease  of  efficiency  of  the  synchronous 
motors. 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  85 

It  is  seen  that  the  curve  of  the  saving  which  results  from  the  use  of 
synchronous  motors  working  jointly  with  induction-motors,  rises  rapidly, 
and  that  it  becomes  positive  for  values  of  cos  <£  which  are  all  the 
greater  the  better  the  efficiency  of  the  synchronous  motors. 

Regulation  of  Distribution-Voltage.  The  over-excitation  of  syn- 
chronous motors  can,  incidentally,  enable  the  voltage  to  be  raised 


at  the  terminals  of  motors  and  transformers  connected  with  the  line, 
and  even  at  the  terminals  of  the  generator.  The  possibility  of  this 
is  readily  seen  in  the  case  where  the  line  has  a  negligible  resistance  and 
reactance,  and  where  there  is  only  one  generator  and  one  motor 
which  are  both  alike.  The  difference  of  potential  at  the  terminals 
(Fig.  41),  is  OB  (B  being  at  the  middle  of  AiA2),  and  by  increasing 


Inferna/  e./n.f  of  genera  for 

FIG.  42. 


CM  2  this  difference  of  potential  can  always   be  made    as    large    as 
desired. 

In  the  general  case,  by  drawing,  successively,  with  their  phases, 
three  vectors  equal  to  the  products  of  the  current  A\C  by  the  impedances 
of  the  generator,  of  the  line,  and  of  the  motor,  we  obtain  a  broken  line 
A\BDA<i  (Fig.  42).  The  voltages  at  the  terminals  of  the  generator 
and  of  the  motor  are  OB  and  OD\  and  A2  can  always  be  sufficiently 
increased  so  as  to  make  OD  and  OB  greater  than  OA\. 


86 


SYNCHRONOUS  MOTORS 


Lahmeyer  made  an  interesting  application  of  this  peculiarity 
at  Bockenheim,  where  polyphase  converters  were  operated  like  true 
synchronous  motors.  By  increasing  their  excitation,  the  induced 
E.M.F.  can  be  raised  in  such  a  way  as  to  obtain  a  secondary  voltage 
of  distribution  which  is  constant  at  all  loads.  The  regulation  can 
be  made  by  hand  or  automatically.  From  this  point  of  view  rotary 
converters  present  a  distinct  advantage  over  stationary  transformers, 
which  introduce  lags  and  whose  secondary  voltage  necessarily  decreases 
with  the  load. 

(2)  Compensation  with  Respect  to  the  Generators.  It  is  a  two- 
fold disadvantage,  for  the  generators  at  the  power  station,  to  have 
the  current  out  of  phase,  first,  because  the  total  current  is  increased 


Z580    C 


Vo/k 


500              IOOO                                ZOOO 
Amperes 


3000 


FIG.  43. 

and,  secondly,  because  the  armature-reaction  is  greater.  To  demon- 
strate this  beyond  question,  let  us  consider  a  generator  supplying  a 
distribution-system  at  constant  potential  U.  Let  Jw  and  Jd  be  the 
active  and  reactive  currents  supplied  to  that  system,  /  being  the 
resultant  current.  Let  RI,  X\,  Zi,  respectively,  be  the  resistance, 
reactance  and  impedance  of  the  generator  and  of  the  feeders;  and 

let  them  be  considered  constant,  and  as  if  located  in  a  single  apparatus. 

•g 
Let  tan  n  =  JT  ^e  tne   corresponding    reactance-factor.     Let    us 

construct  a  diagram  (Fig.  43)  starting  from  the  distribution-voltage 
U,  which  is  to  be  maintained  constant  at  all  loads. 

The  necessary  E.M.F.  E%  will  be  obtained  by  compounding,  with  the 
voltage  OA  =  U,  two  vectors  representing  the  volts  absorbed  by  the 
impedance  under  the  action  of  the  two  currents  Jw  and  /<*.  The 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  87 


first    AAi=ZiJio,  makes,  it  is  known,  the  angle  fi  with  the  voltage 


U.     The  second,  A\A\—Z\J^  will  lag—  behind  the  first,  since  Jd  is 

lagging  —  behind  Jw.     The   resultant    A\A  corresponds  to    the   real 
current 


(33) 


and  the  .line  OA\  represents  the  internal  E.M.F.  which  the  generator 
must  produce  in  order  that  the  distributing  voltage  shall  equal  U. 
The  diagram  gives  the  following  formula,  which  is  analogous  to 
formula  (26): 


nri).  .     .     (34) 

In  this  case  ^  being  always  very  large,  A\A^  is  almost  parallel  to  U. 
In  general,  we  can  write,  approximately, 


(35) 


It  is  seen  that  the  reactive  current  requires  a  much  greater  increase 
of  E:M.F.  than  the  active  current.  If  we  had  Jd=o,  the  alternator 
would  develop  the  same  power  with  a  lower  current,  AAi,  and  a 
lower  E.M.F.,  O4?- 

Let  us  suppose  the  current  /  to  be  the  maximum  which  the  alterna- 
tors can  produce;  and  let  us  describe,  around  A  as  a  center,  a  circle 
of  radius  A  A  \  =  ^\J-  The  manufacturer  must  have  allowed  a  suf- 
ficient margin  in  the  excitation  to  attain  the  E.M.F.  EQ,  which  enables 
this  current  to  be  produced  through  a  dead  resistance  while  keeping 
the  distribution-voltage  at  the  value  Uj  but  if  he  confined  himself  to 
doing  that,  as  was  his  right,  it  would  be  necessary,  when  the  lag 
increases,  to  reduce  the  output  in  such  manner  that  the  point  A\  will 
remain  on  the  circle  AQ€,  described  from  O  as  a  center,  with  EQ  as 
radius. 

In  such  cases  the  effect  of  the  reactive  current  becomes  absolutely 
objectionable,  and  it  is  then  very  necessary  to  reduce  it  by  employing 
means  such  as  those  just  considered. 

Fig.  43  can  serve  as  a  numerical  example  in  the  case  of  a  2000- 
volt  distribution  where  the  receiving  apparatus  has  a  mean  power- 
factor  of  0.70  and  the  generator  has  an  impedance  of  15  ohms  (the 


88  SYNCHRONOUS  MOTORS 

two  scales  being  therefore  in  the  ratio  of  15  to  i).  We  will  now  study 
this  example  in  detail. 

Numerical  Example.  Suppose  a  case  where  energy  is  to  be  dis- 
tributed for  mechanical  purposes  by  means  of  a  line  supplying  a  great 
number  of  induction-motors.  During  the  day,  when  a  portion  of  these 
motors  are  running  at  light  load,  their  mean  power-factor  is  about 
0.70,  i.e.,  the  active  current  Jw  is  approximately  equal  to  the  reactive 
current.  Let  us  assume  that  it  has  been  decided  that  the  total  power 
thus  distributed  will  be  about  100  kw. 

The  distribution-voltage  is  2000  volts;  the  line-current  has  been 
predetermined  on  the  assumption  of  a  power-factor  of  0.70;  and,  there- 
fore, assuming  an  average  efficiency  of  0.75,  the  current  would  be 

T  10.0000 

/=- — -  —=95  amperes. 

2000X0.75X0.70 

The  line,  having  one  ohm  of  resistance  (its  efficiency  being  about 
95  per  cent),  and  being  supposed  to  have  negligible  inductance, 
gives  a  voltage-drop  of  100  volts.  An  alternator  has  therefore  been 
provided,  of  sufficient  capacity  to  produce  95X2100=200  k.w.,  and 
it  is  believed  that  ample  allowance  has  been  made.  This  alternator 
will  have,  for  example,  A"i=i8;  RI  =1.6;  whence  £1=13  ohms, 
and  tan  7-1  =  8.3.  The  machine  furnished  by  the  manufacturer  is 
regulated  for  a  dead-resistance  load  and,  consequently,  it  is  capable 
of  giving,  with  its  fields  almost  saturated,  the  E.M.F.  which  seems 
necessary  for  the  full  load,  by  making  OA=U=2ooo  volts,  and  EQ= 
0^0=2580  volts. 

On  putting  the  machine  in  operation  an  unexpected  drop  of  voltage 
is  noted,  and  it  is  impossible  to  obtain  the  95  amperes  calculated. 
In  fact,  if  we  draw  the  segment  A  A  \  at  an  angle  of  45°,  starting  from 
OA\y  it  is  not  2580  volts  which  would  be  necessary,  but  practically, 


EI  =  OA  i  =  3  230  volts. 

Since  the  machine  can  only  produce  2580  volts,  the  load-point 
will  have  to  be  moved  to  the  point  of  intersection,  a\t  of  the  angle- 
line  7-,  with  the  circle  AQ€.  The  machine  cannot,  therefore,  deliver 
more  than  45  amperes  at  the  voltage  of  2100  volts;  and,  since  the 
corresponding  active  current  is  only  32  amperes,  it  will  be  seen  that 
the  alternator,  although  bought  for  200  k.w.  and  expected  to  be  good 
for  100  k.w.,  can  give,  in  actual  service,  only  21  +  2100=46  k.w.; 
and  yet  the  manufacturer  is  not  to  blame. 


DETAILED  STUDY  OF  OPERATION  WITH  NORMAL  LOAD  89 

To  overcome  the  difficulty,  it  will  be  sufficient  to  move  the  load- 
point  to  ai  ,  because  the  power  will  be  that  needed  (the  active  current 
being  A\ai}  without  requiring  that  the  total  current,  Aa'\,  should 
materially  exceed  the  predetermined  current.  This  only  requires 
that  49  amperes  of  reactive  current  should  be  eliminated  from  the 
67  which  exist.  The  most  advantageous  method  of  obtaining  this 
result  consists  in  replacing  a  few  induction-motors  by  synchronous 
motors.  If  the  change  is  made,  for  example,  with  a  certain  number 
of  motors  representing  0.75X2100  voltsX25  amperes=39.5  k.w.  of 
mechanical  power,  the  reactive  current  suppressed  is  already  25  amperes; 
and  it  is  sufficient,  then,  to  produce  an  equal  current  in  the  opposite 
direction,  by  means  of  synchronous  motors.  If  we  assume  the  latter 
to  have  the  same  efficiency,  they  will  also  absorb  25  amperes  of  active 
current,  so  that  their  total  current  will  be  about  35  amperes  (same  as 
that  of  the  induction-motors  which  they  replace. 

Their  E.M.F.,  calculated,  in  the  manner  above  indicated,  with 
the  object  of  producing  the  25  amperes  of  reactive  current,  will  be 
only  2660  volts,  if  a  single  motor  of  35  amperes  be  used,  and  2525 
volts  if  two  motors  of  17.5  amperes  each  are  used,  etc.  These  voltages 
can  be  attained,  as  a  rule,  with  motors  bought  for  running  at  2000 
volts. 

These  motors  will  necessarily  be  working  under  imperfect  condi- 
tions, since  they  will  be  only  partly  loaded.  The  installation,  as  a 

35 — 25 
whole,   will,   therefore,  be  losing  an  output  of—       -  X  40  =  16  k.w. 

2S 
But,  on    the  other  hand,  it  will  be  gaining,  in  the  generator-output, 

(67  —  31)2100=76  k.w.;  hence  there  should  be  no  hesitation  in 
adopting  this  method. 

The  conditions  taken,  in  this  example,  are  not  exaggerated,  con- 
sidering the  apparatus  which  has  been  used  until  recently,  and  in  which 
a  drop  of  voltage  of  30  per  cent  in  the  alternators  under  full  load  on 
a  dead  resistance  was  a  frequent  occurrence.  These  conditions 
occurred  at  Bockenheim,  where,  according  to  Lahmeyer,  the  output 
of  the  generators  was  doubled  by  introducing  synchronous  motors. 
These  conditions  would  occur  on  many  of  the  actual  distribution  sys- 
tems if  induction-motors  were  extensively  introduced. 

With  the  more  recent  types  of  alternators  the  conditions  are  better, 
and,  consequently,  the  use  of  synchronous  motors  may  be  less  advan- 
tageous. Nevertheless  the  reduction  of  generator-output  by  the  use 
of  synchronous  motors  can  still,  easily,  amount  to  as  much  as  25  to 


90  SYNCHRONOUS  MOTORS 

30  per  cent.  For  this  reason  it  is  always  desirable,  in  any  extensive 
installation,  to  consider  the  expediency  of  utilizing  a  sufficient  number 
of  synchronous  motors,  to  avoid  this  reduction  of  output. 

Comparison  between  Synchronous  and  Induction-Motors.  The 
preceding  remarks  show  the  great  advantages  which  synchronous 
motors  present  in  comparison  with  induction-motors,  in  certain  cases. 
Their  stability  of  operation  is  not  inferior  to  that  of  induction-motors, 
because  a  single-phase  induction-motor,  and  even  a  polyphase  induc- 
tion-motor, also  falls  out  of  step  when  too  much  overloaded,  unless 
the  secondary  resistance  has  been  made  quite  large,  at  the  expense  of 
efficiency.  The  two  points  of  inferiority  of  synchronous  motors  are, 
first,  the  fact  that  they  are  not  self-starting,  and,  second,  their  low 
efficiency.  In  reality,  their  inferiority  in  these  two  respects  is  not 
as  great  as  supposed,  because  it  is  possible  to  make  synchronous  motors 
which  are  self-starting  without  load,  and  as  for  their  efficiency,  it  can  be 
made  as  high  as  that  of  induction-motors,  if  they  are  properly  designed. 

On  comparing  a  synchronous  motor  of  Kapp  type,  with  non- 
laminated  poles  and  expensive  excitation,  consuming,  when  running 
without  load,  a  current  equal  to  one-third  the  full-load  current,  with 
an  induction-motor  of  the  most  perfect  Oerlikon  type  (see  article  of 
Kolben,  Elektrotechnische  Zeitschrift,  Nov.  i,  1894)  it  is  not  difficult 
to  establish  the  superiority  of  the  latter.  BuL,  if,  in  the  said  induction- 
motor,  the  bars  of  the  rotor  were  to  be  transformed  into  a  D.C.  exciting 
circuit,  it  would  be  observed  that  the  difference  in  efficiency  becomes 
very  small. 

Synchronous  motors  have  a  great  advantage  in  construction  because 
they  admit  of  much  higher  magnetic  densities  than  induction-motors. 

[Use  of  Synchronous  Motors  to  Raise  Power-Factor  in  America. 
The  compensating  action  of  over-excited  synchronous  motors  is 
utilized  extensively  in  America  for  improving  the  power-factor  of 
distribution-systems. 

As  an  interesting  example  of  the  practical  application  of  synchro- 
nous condensers  some  figures  obtained  from  the  Detroit  Edison 
Company  are  given  in  Appendix  C.  C.  O.  M.] 


CHAPTER  III 

ADDITIONS  TO  THE  THEORY.     SECOND  APPROXIMATION 

Imperfections  of  the  Theory.  .  Owing  to  the  hypotheses  on  which 
it  is  based  the  preceding  elementary  theory,  in  common  with  all  theo- 
ries concerning  alternating  current  machines,  has  certain  imperfections. 
We  shall  discuss  these  briefly. 

(1)  When  the  currents  are  not  sinusoidal,  the  results  may  be  mate- 
rially modified.     At  this  point,  it  may  be  stated  that  we  should  pre- 
cisely endeavor  to  make  machines  giving  sinusoidal  currents  for  power 
transmission,1  because,  if  their  E.M.F.  do  not  comprise  exactly  the 
same  harmonics,  the  circuit  becomes  the  seat  of  parasite  currents, 
which  are  often  very  important.     For  example  a  certain  Labour  syn- 
chronous motor  having  a  sinusoidal  E.M.F.  when  run  without  load, 
being  supplied  with  current  from  the  distribution  system  of  the  Champs 
Elysees  district,  where  the  E.M.F.  was  non-sinusoidal,  required  a  cur- 
rent almost  double  that  which  it  consumed  at  the  factory  when  supplied 
from  an  appropriate  generator.     This  fact,  among  many  others,  shows 
that  the  effect  of  harmonics  cannot,  by  any  means,  be  considered 
negligible. 

(2)  The  reactance  X  of  the  motor  is  not  constant,  but  varies  with 
the  excitation,  with  the  strength  of  the  armature-current,  and  with  its 
phase-difference  with  respect  to  the  E.M.F.  JL%. 

In  practice,  the  angle  7-  differs  sufficiently  from  90°  to  make  its 
variations,  under  the  influence  of  the  preceding  effects,  very  small. 
Consequently,  the  line  of  zero  phase-difference,  A\N,  in  Fig.  27,  is  a 
curve  which  is  always  sufficiently  flattened  to  warrant  our  continuing, 
without  material  error,  to  compare  it  to  a  straight  line.  But  the  power- 

1  This  question  was  discussed  by  various  authors  in  the  Electrical  World, 
which  had  the  happy  idea  of  opening  its  columns  to  that  very  interesting  con- 
troversy. The  greater  portion  of  those  who  expressed  their  opinion,  and  among 
them  the  author  of  this  work,  were  in  accord  in  recognizing  the  important 
advantages  of  a  sinusoidal  E.M.F.  from  various  points  of  view 

91 


92  SYNCHRONOUS  MOTORS 

circles  will  be  more  or  less  deformed  by  these  variations  of  reactance. 
The  saturation  of  the  fields,-  for  strong  excitation-currents,  reduces 
these  deformations  and  raises  the  right-hand  branch  of  the  V-curves 
more  rapidly  toward  the  top  than  calculation  would  indicate.  It  is 
generally  admitted  that  there  is  no  other  way  of  taking  these  deforma- 
tions into  account  than  by  plotting  the  V-curves  experimentally. 

(3)  No  attention  has  been  paid  to  the  variation  of  losses  in  the 
armature  other  than  the  resistance-loss.  It  may  be  remarked  that  these 
losses  will  vary  but  little,  since  the  friction  is  substantially  constant,  and, 
in  a  motor  supplied  from  a  constant  potential,  the  resultant  E.M.F.  in 
the  motor  remains  almost  constant.  Likewise  the  induction  and  the 
hysteresis  losses  and  the  eddy  currents  remain  approximately  constant. 

The  only  variable  losses  are  those  due  to  eddy  currents  in  the 
fields,  and  the  energy  expended  for  excitation,  neither  of  which  have 
been  taken  into  consideration  in  what  precedes. 

Practically,  therefore,  no  great  error  has  been  made  in  supposing 
the  losses  in  the  iron  to  be  constant  while  increasing  slightly  the  no- 
load  losses  (by  way  of  compensation). 

Attention  is,  therefore,  called  to  this  last  source  of  imperfection 
merely  to  show  its  influence.  In  reality,  the  same  thing  might  also  be 
said  in  regard  to  the  first  two  causes  of  imperfection,  because  the 
true  object  of  this  theory  is  only  to  elucidate  the  general  course  of  the 
phenomena,  and  to  give  a  method  of  calculation  which  will  be  sufficient 
for  practical  purposes.  However,  the  effects  to  which  attention  is 
called  above  may  assume  sufficient  importance,  at  times,  to  justify  a 
somewhat  more  detailed  investigation.  This  is  more  especially  true  in 
regard  to  variations  of  reactance. 

Variations  of  Reactance  with  Lag  of  Current  and  Saturation  of 
Fields.  Armature  -Reaction.  The  author  has  shown,  elsewhere  (L>  In- 
dustrie Electrique,  1899,  p.  481)  that  the  self-induction  of  any  alternator 
can  be  represented  by  two  armature-reactions:1 

(i)  An  armature-reaction  directly  opposed  to  the  exciting  ampere- 
turns  and  which  is  equivalent  to 

KN'Id 


counter-ampere-turns.  In  this  equation  7d=the  effective  amperes 
of  reactive  current,  with  respect  to  E2  ;  Nf  equals  the  number  of  arma  • 
ture  conductors  in  the  induced  field;  and  K  equals  a  coefficient  which 

1  See  Part  II,  Chapter  I. 


ADDITIONS  TO  THE  THEORY.     SECOND  APPLICATION    93 

depends  on  the  shape  of  the  pole-pieces.  The  induction-flux  is  pro- 
duced by  the  algebraical  resultant  of  the  exciting  ampere-turns  and 
of  these  counter-ampere-turns. 

(2)  A  transverse  reaction  producing  a  flux  which  closes  itself  trans- 
versely in  the  pole-pieces,  without  penetrating  the  field-windings 
and  having  a  numerical  value  equal  to 


wherein  L'=a  coefficient. 

This  resolution,  of  which  the  validity  has  been  shown  else- 
where, takes  into  account  the  lag,  since  the  current  intervenes  by 
its  two  components  (active  and  reactive)  with  respect  to  the  E.M.F. 
of  the  motor.  [The  words  "  active  "  and  "reactive"  are  here  taken 
relatively  to  an  E.M.F.  induced  with  open  circuit.  The  phase-angle 
thus  obtained  is  therefore  slightly  different  from  that  which  would  be 
found  at  the  terminals.]  It  also  takes  into  account  the  saturation 
of  the  fields,  since  the  resultant  flux  of  the  fields  is  calculated  from 
the  ampere-turns  themselves,  by  reference  to  an  excitation-curve 
obtained  experimentally. 

As  to  the  transverse  flux,  it  is  substantially  independent  of  the  satura- 
tion, because  the  reluctance  of  the  magnetic  circuit  through  which 
it  finds  its  path  is  substantially  constant.  That  is  why  JJ  can  be  con- 
sidered constant. 

With  this  assumption,  it  is  easy  to  establish  a  new  diagram  for 
synchronous  motors  which  will  take  these  reactions  into  account.  It 
will  be  sufficient  to  segregate  the  motor-reactances.  Let  z  and  x  equal 
the  impedance  and  the  reactance,  respectively,  of  the  generator  and 
of  the  line,  and  of  the  magnetic  leakage  of  the  motor;  and  let  R  equal 
the  resistance  of  the  circuit,  including  the  motor-resistance.  (The 
generator-impedance  may  be  neglected  when  the  current  is  taken 
from  a  distribution-system.)  Let  £2  equal  the  effective  motor  E.M.F., 
i.e.,  the  E.M.F.  produced  by  the  resultant  fields  of  the  field  and 
armature  ampere-turns. 

The  generator  E.M.F.  EI  will  be  equal  to  the  resultant  of  the  following 
three  components:  First,  the  effective  counter  E.M.F.,  A^O=t^  with 
its  sign  reversed;  second,  the  transverse  reaction  —  E.M.F.  A2B=^LfIw 


with  its  sign  reversed  and  having  a  phase-lead  of  —  ahead  of  £2]  and, 


94 


SYNCHRONOUS  MOTORS 


third,  all  the  load-losses  produced  by  the  impedance  z;  this  latter 
being  represented  by  a  vector  BAi  =  zI  (Fig.  44)  which  makes  the 
angle  0  with  a  line  BD  that  is  itself  drawn  at  an  angle  d  with  OA% 
such  that 


The  line  BA  i  can  serve  to  define  the  current  I,  in  magnitude  and 
in  phase,  if  B  Y  be  taken  as  the  axis  of  reference,  with  a  scale  Z  times 
greater  than  the  scale,  of  volts. 


This  vector  zl  can  be  resolved  into  two  E.M.F.'s,  viz.: 
BD=zIw,  produced  by  the  active  current,  and  DA  =  zId,  produced  by 
the  reactive  current. 

The  line  A2D,  closing  the  triangle  A2BD,  then  represents  the 
combination  of  the  reaction  — E.M.F.'s  due  to  the  active  current,  and  it 
makes,  with  OX,  the  angle  f  defined  by  the  equation 

/  T  f     I      7\ 

tan  r=  M 


R 

[It  will  be  noted  that  the  diagram  differs  from  the  preceding  by  the 
definitions  of  z  and  d,  which  do  not  include  the  motor-reactance.] 
Moreover,  we  will  have 


ADDITIONS  TO  THE  THEORY.     SECOND  APPLICATION    95 

wherein  Ni  equals  the  exciting  ampere-turns,  and /equals  the  function 
which  defines  the  law  of  variation  of  the  induced  E.M.F.  with  the 
total  inducing  ampere-turns  (this  law  being  defined,  in  practice,  by 
means  of  a  curve).  The  ampere-turns  of  the  armature  are,  in  fact, 
demagnetizing,  whenever  Id  lags^behind  £2,  i-e.,  when  the  point  A\ 
is  at  the  right  of  the  axis  BY',  and  they  are,  on  the  contrary,  magnetiz- 
ing, when  AI  is  at  the  left. 

The  segments  zlw  and  zld  again  measure,  in  magnitude  and  in 
phase,  the  active  and  reactive  amperes  by  reference  to  some  suitable 
scale;  and  the  phase-angles  (lags)  are  measured  from  the  line  BY, 
which  serves  as  axis  of  reference. 

The  output  is  the  sum  of  the  outputs  corresponding  to  the  currents 
Iw  and  Id.  The  induced  E.M.F.  is  composed  of  the  E.M.F.  £2,  pro- 
duced by  the  direct  field,  and  of  the  E.M.F.  (ajLfIw),  produced  by  the 
field  of  transverse  reaction.  The  first  is  in  phase  with  Iw  and  the  second 
with  Id.  We  therefore  have 

P2  =  £2!™ — toL'IwId' 

The  diagram  therefore  gives  us  again  all  the  conditions  of  operation 
of  the  motor  without  material  increase  in  complication.1 

First  Application  of  Corrected  Diagram.  Determination  of 
Reactive  Current  as  a  Function  of  the  Excitation,  with  Constant 
Active  Current.  The  preceding  diagram  enables  us  easily  to  find 
the  values  of  the  reactive  current  as  a  function  of  the  excitation  and  of 
the  active  current,  by  the  additional  assistance  of  the  excitation-curve 
for  open  circuit,  either  obtained  experimentally  or  else  predetermined 
by  calculation.  This  curve  (Fig.  45)  gives,  for  each  value,  F,  of  the 
ampere-turns,  the  corresponding  induced  E.M.F. 

'The  problem  is  really  much  more  complex,  especially  when  the  load 
changes  suddenly,  since  the  effect  on  the  field-flux  due  to  the  armature-reac- 
tion does  not  then  appear  as  fast  as  the  change  of  armature-current.  The 
flux  may  indeed  remain  constant  for  several  seconds,  even  if  the  armature-cur- 
rent changes  considerably.  Hence,  for  sudden  changes  of  load  the  phenomenon  is 
largely  governed  by  the  true  self-induction  of  the  armature  only.  Thus,  the 
armature-reactance  may  be  only  from  i  per  cent  to  3  per  cent,  whereas,  for 
slowly  changing  loads,  it  would  be  equal  to  the  synchronous  reactance,  say, 
40  per  cent.  Therefore,  machines  which  have  good  mutual  induction  between 
field  and  armature,  such  as  high-speed  motors  and  turbogenerators,  may  be 
unstable  if  the  line-loss  is  great  and  if  the  load  changes  suddenly.  The 
remedy  in  such  cases  is  to  install  external  reactances. — E.  J.  B. 


96  SYNCHRONOUS  MOTORS 

Let,  for  example,  Iw  equal   the  given  active  current.      Knowing 
the  angle  d  (Fig.  46),  the  segment  A2B=ajL'Iw  perpendicular  to  the 


axis  A2X  can  be  drawn,  and  then  the  segment  BD=zIw  making  the 
angle  d  with  A2X  can  be  also  drawn. 


ADDITIONS  TO  THE  THEOKY.     SECOND  APPLICATION    97 

We  thus  obtain  the  line  of  reference  BY,  which  represents  the 
condition  when  the  current  is  in  phase  with  respect  to  £2-  The  locus 
of  the  characteristic  point  A\  is  a  straight  line  DQ  perpendicular  to 
BD,  while  the  locus  of  the  point  O\  is  the  straight  line  OX.  For 
each  value  /„,,  we  therefore  know  the  two  straight  lines  at  which  the 
vector  AiOi,  equal  to  the  supply  E.M.F.  E\,  should  always  terminate. 

All  loads  corresponding  to  the  values  Iw  will  therefore  be  determined 
by  taking  points  such  as  Ai,Ai',A"i,  etc.,  on  DQ,  and  by  drawing  circu- 
lar arcs  of  radius  E±.  The  respective  intersections  Oi,Oi',Oi",  of  these 
lines  with  O\A^  give  some  E.M.F.'s  values,  £2,  which  are  proportional 
to  A2Oi,  A2O\,  A2Oi",  while  the  corresponding  reactive  currents, 
Id,  are  proportional,  respectively,  to  the  lengths  DAi,  DAi',  DA\" . 

For  each  pair  of  values  of  Id  and  £2,  the  corresponding  exciting 
ampere-turns  F  will  then  be  obtained  by  referring  to  the  excitation- 
curve.  Let  us  take,  on  this  curve,  the  point  M,  which  has  for  its 
ordinate  the  resultant  E.M.F.  e-2,  thus  determined.  The  correspond- 
ing abscissa,  Om,  measures  the  resultant  ampere-turns. 

F±-±=  KN'Id. 

V2 

It  is  therefore  sufficient  to  subtract,  from  this  abscissa,  the  counter- 
ampere-turns  of  the  armature, =KN'Id,  in  order  to  obtain  the  abscissa, 
Qm'=  F,  which  represents  the  excitation,  and,  consequently,  represents 
E2,  the  E.M.F.  induced  on  open  circuit  which  would  be  obtained  with 
this  excitation. 

It  must  not  be  forgotten  that  the  reactive  current  Id  'is  demagnetizing 
and  positive  if  counted  to  the  right  of  the  axis  YY',  and  magnetiz- 
ing and  negative  if  counted  to  the  left  (see  page  94).  The  upper 
signs  in  the  above  formulae  apply  in  the  first  case,  and  the  lower  signs 
in  the  second  case.  We  will  therefore  have  E2  <  £2  if  the  point  A\ 
is  to  the  right  of  the  point  D  and,  on  the  contrary,  E2  >  £2  if  it  is  to 
the  left. 

In  the  particular  case  where  a  motor  is  connected  directly,  without 
additional  rheostat  or  self-induction,  on  a  constant  potential  circuit 
of  great  output,  and,  when  the  impedance  z  of  its  leakage  is  negligible, 
D  coincides  with  B,  and  the  line  QD  then  passes  through  the  point 
B  itself;  which  simplifies  the  diagram  still  further. 

The  diagram  which  has  just  been  explained  is  most  precise  in  the 
case  of  an  alternator  working  at  the  knee,  or  above  the  knee  of  the 


98 


SYNCHRONOUS  MOTORS 


characteristic  curve,  or  presenting  unequal  coefficients  of  direct  and 
transverse  reactions.  In  the  elementary  theory  it  was  assumed,  on  the 
contrary,  that  the  permeability  of  the  field-circuit  is  practically  constant 
and  that  the  two  reaction-coefficients  are  equal. 

Particular  Case  where  the  Permeability  of  the  Field-Circuit  is 
Constant  and  the  Two  Reaction-Coefficients  are  Equal.  It  is  inter- 
esting to  note  how,  in  this  case,  the  new  and  more  general  diagram  com- 
prises the  previous  forms  as  a  particular  solution.  It  is  only  necessary, 
indeed,  to  observe  that  the  coefficient  of  transverse  reaction,  L',  then 


FIG.  47. 

becomes  equal  to  what  we  have  termed  the  coefficient  of  mean  self- 
induction  L.  Likewise,  the  direct  armature-reaction,  which  produces 

—  =  KNfId  counter-amperes-turns  due  to  "the  reactive  current,  gives  rise 

V2 

to  a  counter  E.M.F.  which  can  be  represented  by  wLId,  Therefore, 
between  the  effective  E.M.F.,  £2,  and  the  open  circuit  E.M.F.  obtained 
with  the  same  excitation,  we  have  the  relation 


Id  being  always  taken  as  positive  on  the  right-hand  side  of  BY  and  as 
negative  on  the  left-hand  side.    Suppose,  for  example,  Id  to  be  negative. 


ADDITIONS  TO  THE  THEORY.     SECOND  APPLICATION    99 

It  would  be  necessary,  in  order  to  obtain  E2)  to  add  to  the  segment 
OiA2  (Fig.  47),  a  section 


It  amounts  to  the  same  thing  if  this  segment  is  drawn  parallel  to 
and  to  the  left  of  AI,  along  A\A'\,  and  if  a  circular  arc,  of  radius  E\, 
is  drawn  from  A'2,  to  cut  OA2  precisely  at  the  point  O'\. 

Now  if  we  join  DA'i,  it  is  seen  that  the  triangles  DA^A\  and 
DA2B  are  "  similar  "  triangles,  since  the  side  DA\  and  DB,  also 
A\A\  and  BA2  are  respectively  perpendicular  and  proportional  each 
to  each.  It  may  be  deduced  from  this  that  the  line  DA'\  is  per- 


FIG.  48. 

pendicular  to  DA2.  On  the  other  hand,  the  vectors  A2D  and  DA'i 
being  the  resultants  of  the  loss  of  potential  in  the  motor  and  in  the  rest 
of  the  circuit,  evidently  represent  ZIW  and  Z/d,  respectively,  when  Z 
equals  the  total  impedance,  of  the  circuit.  The  diagram  composed  of 
the  right  lines  A2D,  DA\,  AiA\,  A2O,  is,  therefore,  nothing  more 
than  the  diagram  which  would  have  been  obtained  directly  by  the  first 
method.  Q.E.D. 

Fig.  48  shows  the  same  analysis  for  the  case  where  Id  is  lagging 
(Id  being  positive).  The  precision  of  the  power-formula  already  given 
can  also  be  verified;  for,  in  this  formula, 


P=  eIw— 


100  SYNCHRONOUS  MOTORS 

if  we  substitute  for  s2>  its  value, 


we  have 

P2=/2/u>, 

as  before. 

Second  Application.  Operation  with  Constant  Excitation,  on 
Constant-Potential-Supply  System.  When  once  the  V-curves  have 
been  drawn  it  is  very  easy  to  obtain  from  them  the  different  load- 


V 


FIG.  49. 


values  corresponding  to  any  given  constant  excitation.  These  values 
can  also  be  determined  directly  from  the  diagram  in  the  following 
manner: 

Let  us  take  the  reactive  current  Id  and  let  us  find  the  active  current 
Iw  corresponding  to  each  value  thereof.  The  direction  A2D  (Fig.  49) 
being,  therefore,  constant,  can  be  drawn  once  for  all.  Likewise,  the 
direction  A2W,  to  which  the  reactive  segments  DAi  are  parallel,  makes 


ADDITIONS  TO  THE  THEORY.     SECOND  APPLICATION  101 


with  OA2  a  known  angle, d.      Since  we  assume  the  values  of  7d, 

we  can  calculate  the  values  zld  and  — -  KN'Id.     (As  before  z  represents 

V2 

the  impedance  of  the  armature  only). 

Let  us  draw,  on  A2W,  a  segment  A2A  =  zId.  The  point  A  must 
fall  on  a  right  line,  AA\,  which  passes  through  the  point  A  and  which 
is  parallel  to  A2D.  Again,  knowing  E2,  the  excitation-curve  gives 
the  excitation  ampere-turns  F.  Subtracting  %KN'Id,  we  have  the 
resultant  ampere-turns,  and  consequently,  the  effective  E.M.F.  e2. 
Let  this  be  drawn  on  A2X,  to  A2O.  The  load-point  AI  must  fall  on 
a  circle  described  around  O  as  a  center  with  E\  as  radius.  It  is 
therefore  determined  by  the  intersection  of  that  circle  with  the  right 
line  AA\  previously  drawn. 

We  thus  obtain  the  value  of  z!w=±BD,  and,  consequently,  the 
current  7W.  From  this  we  deduce  the  electric  power  applied  to  the 
motor, 

P2=e2fw-ajL'Iwyjd. 


On  the  other  hand,  I=\/Id2+Iw2.  We  therefore  obtain,  by  points, 
the  relations  between  7  and  P2,  both  expressed  as  a  function  of  e2 
which  serves  as  an  intermediary  variable. 

V-Curves.  The  diagram  does  not  lend  itself  to  the  direct  determina- 
tion of  the  V-curves  for  constant  power,  owing  to  the  complicated 
expression,  which  is  not  explicit  as  a  function  of  E2  and  7W,  as  it  was  in 
the  elementary  theory. 

But  these  curves  can  be  deduced  from  the  curves  of  constant  excita- 
tion, when  these  are  drawn  by  the  process  indicated  in  the  preceding 
paragraph. 

Influence  of  Field-Saturation  on  Stability.  The  stability  is  evidently 
greater  the  higher  the  E.M.F.  of  the  motor  at  the  limit  of  lag.  Now,  for 
the  same  reactive  current,  the  loss  of  potential  produced  by  the  direct 
reactance  is  all  the  less  the  higher  the  degree  of  excitation  above  the 
knee  of  the  characteristic.  It  would  therefore  be  advantageous,  from 
the  standpoint  of  stability  alone,  to  have  the  fields  saturated.  But 
there  is  a.  limit,  here,  owing  to  the  amount  of  energy  required  for  excita- 
tion, and  the  desirability  of  a  certain  margin  to  enable  the  voltage  to 
be  raised. 

The  cut-and-try  process  involved  in  finding  the  co-ordinates  of 
the  V-curves  may  be  obviated,  without  constructing  any  curves,  by 


102  SYNCHRONOUS  MOTORS 

utilizing  Fig.  46.  Knowing  the  value  of  P.  and  assigning  a  value  to 
/„,,  the  equation, 

P2=  £2lw  —  d)L'IwId 

can  be  written  as  follows: 

£2—(t)L'Id=j^=  constant  =M (a) 

1  w 

Taking,  in  Fig.  46,  a  point  AI  on  the  right  line  DQ,  as  a  center, 
let  a  circle  be  described  with  a  radius  equal  to  EI.  Let  O]  be  the 
intersection  of  this  circle  with  the  right  line  A2X.  The  values  of  £2 
and  Id  can  then  be  immediately  deduced.  We  find 

£2= A^O[ 

and  Id=—  DA  i. 

z 

These  values  introduced  in  equation  (a)  will  give,  for  example, 
-  S2-atL'Id=N. 

Should  this  value  be  greater  than  M,  it  is  because  the  point  AI  was 
located  too  near  D.  If,  on  the  contrary,  the  value  N  is  less  than 
M,  it  is  because  the  point  AI  is  too  far  from  D.  The  point  AI  may 
be  moved  until  the  values  found  for  £2  and  Id  give  a  value  for  N  equal 
to  M.  We  can  also  proceed  otherwise.  Having  found,  quite  easily, 
by  the  above  method,  two  positions  such  as  AI  and  A\  which  give, 
for  the  polynomial  N,  two  values,  one  a  little  too  large  (N)  and  the 
other  a  little  too  small  (N"),  a  point,  AI,  can  be  obtained  by 
interpolation  such  that 

'A1/Al       N-M 

Ai'W    M-N"' 

This  gives  the  desired  value  of  the  power,  P,  with  a  sufficient  degree 
of  approximation  for  practical  purposes. 

Influence  of  the  Wave-Form  of  E.M.F.  The  shape  of  the  E.M.F. 
curves  of  the  generator  and  motor  should  approach  the  sinusoidal 
form  as  rhuch  as  possible  in  order  to  avoid  instability  and  low  efficiency. 
Let  us  first  suppose  that  only  one  .of  the  two  machines  departs  from  the 
sinusoidal  form.  By  virtue  of  Fourier's  theorem,  its  E.M.F.  can  be 
considered  as  the  resultant  of  the  superposition,  upon  a  principal 
sinusoidal,  of  higher  harmonics,  i.e.,  of  sinusoidals  of  higher  frequency 
(odd  multiples). 


ADDITIONS  TO  THE  THEORY.     SECOND  APPLICATION   103 

The  fundamental  can  alone  produce  useful  power,  because  it  is 
known  that  energy  cannot  be  produced  by  the  action  of  an  E.M.F. 
until  the  current  has  the  same  frequency.  Since  the  harmonic  terms 
exist  only  in  one  of  the  machines  they  will  simply  produce  parasite 
currents  therein  which  will  increase  the  heating  by  resistance-loss  with- 
out any  useful  result. 

It  is  known  that  the  square  of  the  effective  current  is  equal  to  the 
sum  of  the  squares  of  the  effective  currents  of  the  component  siimsoidals. 
The  heating  produced  separately  by  each  of  the  harmonics  will  there- 
fore be  added  to  the  principal  heating.  This  effect  makes  itself  readily 
apparent  in  the  V-curves  by  an  increase  in  the  minimum  current, 
which  will  be  much  greater  than  would  be  necessary  to  obtain  the  power 
actually  developed.  This  can  easily  be  observed  by  means  of  a  watt- 
meter, which  will  indicate  a  real  power  no  longer  equal  to,  but  really 
lower  than,  the  apparent  power  (product  of  the  effective  current  by  the 
effective  voltage).  It  cannot  be  possible  that  cos  <j)  is  lower  than  unity, 
since  the  current  is  in  phase,  as  we  know,  under  these  conditions. 

It  can  even  happen,  as  noted  by  Bedell  and  Ryan  (Bedell  and  Ryan, 
Journal  of  Franklin  Institute,  Mar.,  1895,  Fig.  9),  on  a  motor  having 
a  strong  third  harmonic,  that  the  V-curve  breaks  up  into  two  branches, 
one  of  which  is  normal,  at  high  excitations,  the  other  being  parasite, 
and  very  high  at  low  excitations.  Between  these  there  is  no  stable 
load  possible,  without  adding  a  strong  self-induction  in  series.  The 
author,  who  has  himself  observed  similar  phenemona  (A.  Blondel. 
Electrotechnische  Zeitschrift,  April  25,  1895),  attributes  this  favorable 
effect  of  a  large  self-induction  (an  effect  which  is  contrary  to  the  theory 
given  on  page  53)  to  the  obliteration  (throttling)  of  the  harmonics, 
which  meet,  in  passing  through  the  self-induction,  a  reactance  that  is 
proportional  to  their  own  frequency. 

In  the  case  where  it  is  the  generator  which  has  a  distorted  E.M.F., 
it  can  be  eliminated,  as  M.  Perot  has  shown  (Perot,  Comptes-Rendus, 
6  Aug.,  1900,  p.  337),  by  connecting  in  parallel,  on  the  line  supplied 
by  this  generator,  a  synchronous  motor  (or  a  converter)  having  a  sinu- 
soidal E.M.F.  This  apparatus,  opposing  no  harmonic  to  those  of  the 
generator,  short-circuits  them  without  reducing  the  principal  sinu- 
soidal. If,  therefore,  we  take  Z  equal  to  the  generator-impedance 
(or  that  of  the  group  of  generators)  and  z  equal  to  the  motor-impe- 
dance, the  amplitude  of  the  harmonics,  at  the  terminals,  will  be 

substantially  reduced  to  •=—  of  their  value  on  open  circuit.     Perot  has 


104  SYNCHRONOUS   MOTORS 

thus  been  able,  in  a  practical  case,  to  reduce  these  harmonic  terms 
to  one-eighth  of  their  value. 

The  case  is  different  when  the  E.M.F.'s  of  the. two  machines  are 
equally  distorted,  because  the  harmonics  which  produce  the  parasite 
current  are  then  resultants  of  harmonics  of  the  same  order  in  the  two 
machines.  Since  these  harmonics  are  all  of  odd  order,  the  two  sets 
will  oppose  each  other,  if  the  principal  sinusoidals  are  themselves 
opposed,  as  is  approximately  the  case  under  normal  operation  when 
there  is  no  lag. 

Consequently,  if  like  machines  are  used  as  generators  and  as  motors, 
a  considerably  distorted  E.M.F.  curve  would  produce  parasite  effects 
which  are  much  less  important  than  in  the  first  case,  and  almost 
negligible  if  care  is  taken  to  have  the  operation  correspond  to  the 
lower  part  of  the  bend  of  the  V-curve. 

Simplified  Diagrams.  The  diagrams  shown  in  Figs.  44,  46,  49 
correspond  to  the  most  general  case  where  it  is  desired  to  take  into 
account  both  the  direct  and  the  transverse  armature-reactions. 

Inasmuch  as  the  direct  armature-reaction  alone  affects  the  total 
excitation  ampere-turns,  it  may  seem,  to  some,  an  unnecessary  com- 
plication to  introduce  the  transverse  reaction  in  the  diagrams.  For 
the  benefit  of  those  who  prefer  Kapp's  simplified  theory,  other 
diagrams  have  been  prepared  (Figs.  440,  46(1,  490)  in  which  the  trans- 
verse reaction  is  neglected,  i.e.,  where  it  is  assumed  that  Iw=o. 

In  constructing  these  simplified  diagrams  it  is  important,  as  before, 
to  determine  the  values  of  the  resultant  E.M.F.  £2  by  reference  to 
the  law  of  magnetic  saturation  as  represented  by  the  E.M.F.  charac- 
teristic curve  shown  in  Fig.  45^.  In  this  diagram  the  abscissae  represent 
excitation  ampere-turns,  and  the  ordinates  represent  E.M.F.'s.  If  the 
point  M'  represents  the  E.M.F.  £2  with  open  circuit,  and  if  Omr 
represents  the  corresponding  excitation  ampere-turns,  then,  on  sub- 
tracting from  Om  the  ampere-turns  corresponding  to  the  reactive 

current  Id  or  making  mm'  =       r      the  ordinate  Mm  corresponding 

V  2 

to  the  abcissa  Om  will  be  the  E.M.F.  £2- 

The  impedance  involved  (z)  is  that  which  corresponds  to  the 
magnetic  leakage  of  the  armature.  Fig.  440  shows  how  the  cor- 
responding potential  difference  zl  can  be  resolved  into  two  components 
obtained  by  projecting  AiA2oi].  YY'  and  on  the  line  DA\  perpendicular 
thereto. 

The  component  DA±  gives  the  value  of  Id  when  2  is  known. 


ADDITIONS  TO  THE  THEORY.     SECOND  APPLICATION  105 

Fig.  460  shows  how  the  values  necessary  for  constructing  the 
F-curves  are  obtained;  and  Fig.  490  represents  the  case  where  the 
synchronous  motor  is  operating  with  constant  excitation  and  at  con- 
stant potential. 


FIG.  46a. 


FIG.  490. 


These  diagrams  give  a  degree  of  precision  which  is  generally 
sufficient  for  practical  purposes,  being  intermediate  between  that 
obtainable  with  the  simplified  methods  used  in  Chapter  II  and  the 
more  rigorous  method  used  in  Chapter  III.  Moreover,  the  diagrams 
440,  460,  490  express  armature-reactions  in  terms  of  magnetomotive 
force  more  precisely  than  if  the  total  zl  resultant  were  to  be  expressed 
by  a  magnetomotive  force. 


CHAPTER  IV 
OPERATION   OF   SYNCHRONOUS  MOTORS.     HUNTING 

Synchronous  motors  are  not  self-starting,  the  same  as  other  electric 
motors,  but  they  have  to  be  started  by  some  artificial  means  such  as 
described  hereinafter. 

Starting  by  Direct  Current.  The  simplest  way,  when  there  is  a 
source  of  direct  current  and  the  synchronous  motors  are  provided  with 
a  commutator  for  producing  direct  current  for  self-excitation,  is  to 
start  the  machine  as  if  it  wrere  a  D.C.  motor,  with  the  armature  and 
fields  connected  in  parallel.  This  plan  can  be  especially  adopted 
in  transformer  stations  using  motor-transformers,  so  as  to  start  a  set 
of  motors  operating  dynamos  as  soon  as  one  has  been  put  in  operation. 

Starting  with  Alternating  Current  by  Polyphase  Motors.  When 
there  is  no  D.C.  source,  the  alternating  currents  themselves  must  be 
used  for  starting  the  motor,  either  by  mounting  on  the  shaft  of  the 
machine  a  small  induction-motor  which  serves  to  run  it  without  load, 
or,  more  frequently,  by  using  the  machine  itself  as  an  induction-motor. 
(When  a  small  motor  is  used  its  capacity  should  be  about  10  per  cent 
of  that  of  the  machine.  It  can  be  even  less,  if  it  can  withstand  over- 
loading for  a  few  seconds.) 

When  using  the  machine  itself  as  an  induction-motor,  the  first 
step  is  to  suppress  the  regular  excitation,  which  would  prevent  the  arma- 
ture from  moving.  The  second  step  is  to  connect  the  armature  with 
the  source  of  alternating  supply,  taking  care  to  lower  the  voltage  suf- 
ficiently to  prevent  excessive  current  through  the  armature.  The 
armature  then  behaves  as  if  it  were  the  primary  winding  of  a  revolving 
field  machine  in  which  the  fields,  and  more  especially  the  pole-pieces, 
play  the  role  of  secondary  circuit.  To  obtain  a  torque  equal  to  a  quarter 
the  normal  torque,  it  usually  requires,  by  this  method,  a  current  at 
least  double  the  normal  current.  To  complete  this  action,  it  is  well 
to  short-circuit  the  field-winding.  Certain  manufacturers  have, 
for  this  purpose,  arranged  in  the  pole-pieces,  as  shown  in  Fig.  50,  a 
series  of  slots  containing  windings  or  bars  of  copper,  connected  together 

106 


OPERATION  OF  SYNCHRONOUS  MOTORS.     HUNTING     107 

like  the  armature  of  an  induction-motor.  In  that  case,  the  motor 
starts  normally,  and  with  a  current  which  depends  on  the  resistance. 
In  any  case  this  current  is  higher  than  in  an  induction-motor  because 
the  air-gap  is  greater  and  requires  a  stronger  magnetizing  current. 
For  this  reason,  it  is  well  to  start  the  motors  without  load,  in  order 
to  avoid  excessive  starting  currents. 

The  production  of  the  revolving  field  results  from  the  eddy  currents, 
but  in  the  absence  of  these,  the  pole- 
pieces,  even  when  laminated,  'give 
rise  to  hysteresis  effects  which  are 
often  sufficient  for  starting  the  motor 
without  load.  In  certain  motors 
having  laminated  poles  the  starting 
current  is  about  double  the  normal 
full-load  current. 

This  effect  is  explained  by  the 
fact  that,  during  the  rotation  of  the 
field,  there  is  a  stronger  attraction  FIG.  ^0. 

in  the  direction  of   motion  than  in 

the  opposite  direction  owing  to  the  lag  of  the  magnetization  behind  the 
changes  of  field  producing  it. 

Synchronism.  In  proportion  as  the  armature-speed  increases, 
the  magnetic  pulsations  produced  by  it  in  the  pole-pieces  become  less 
numerous,  as  can  be  seen  on  connecting  a  lamp  to  the  field-circuit 
and  noting  its  variations  of  brightness.  It  is  well  to  provide  a  centrifugal 
regulator  which  connects  this  lamp  in  circuit  only  when  the  speed 
approaches  synchronism,  because,  at  lower  speeds,  it  would  be  sub- 
jected to  excessive  voltage.  When  synchronism  is  almost  attained — 
(the  speed  generally  remains  slightly  lower,) — the  fields  are  excited 
when  the  phases  come  into  opposition,  the  motor  falls  into  step,  and 
the  current  immediately  diminishes,  owing  to  the  disappearance  of  the 
reactive  current  which  was  absorbed  up  to  that  time.  The  more  care- 
fully the  time  has  been  selected  for  closing  the  excitation-circuit,  the 
more  easily  the  motor  will  fall  into  step.  It  is  well,  as  a  rule,  to  include, 
in  the  circuit,  a  variable  self-inductance,  which  has  the  effect  of  pre- 
venting excess  of  current  and  of  damping  objectionable  harmonics. 

When  once  the  motor  is  in  synchronism,  it  can  be  loaded  progres- 
sively, by  shifting  the  belt  from  the  idler  to  the  driving  pulley.  This 
should  be  done  very  carefully  when  the  mechanical  resistance  (load) 


108  SYNCHRONOUS  MOTORS 

In  the  particular  case  where  it  is  possible  to  start  the  generator  and 
the  motor  at  the  same  time,  the  operation  of  starting  becomes  easier 
and  there  is  no  necessity  of  suppressing  the  excitation  of  the  motor. 
The  motor  starts  of  itself  by  giving  it  a  slight  impulse,  it  falls  into  step 
at  the  very  first  period,  owing  to  the  low  reactance  at  low  speed,  and 
its  motion  then  accelerates  synchronously  with  that  of  the  generator, 
increasing  gradually  to  full  speed. 

Observations  on  the  E.M.F.  Induced  in  the  Poles.  During  the 
starting  period,  the  fields  are  the  seat  of  an  important  alternating 
flux  whose  frequency  attains,  at  first,  the  same  frequency  as  the  alternat- 
ing currents  themselves.  The  result  is  the  production  of  a  very  high 
E.M.F.  in  the  field-winding,  which  is  generally  made  of  fine  wire. 
If,  therefore,  this  winding  remains  on  open  circuit,  its  insulation  should 
be  particulary  good,  and  contact  should  not  be  made  with  its  terminals. 
It  is  preferable  to  short-circuit  this  winding.  This  alternating  E.M.F. 
naturally  vanishes  when  synchronism  is  attained.  It  is  obvious  that 
the  armature-brushes  on  the  D.C.  side  must  also  be  disconnected, 
since,  otherwise,  the  pulsating  current  would  produce  a  braking  effect 
which  would  prevent  the  motor  from  starting. 

Accessory  Starting  Apparatus.  Installation  of  Synchronous 
Motors.  The  starting  process  involves  the  use  of  two  kinds  of  accessory 
apparatus,  i.e.,  phase-indicators  and  current-controllers. 

It  is  also  well  to  provide  fuses  and  automatic  circuit -breakers  which 
break  the  main  circuit  whenever  there  is  no  excitation. 

Phase-Indicators.  Phase-indicators  play  the  same  role  in  this 
case  as  in  coupling  alternators  in  parallel.  They  are  of  the  same  kind 
and  are  connected  in  the  same  way.  Their  use  is  recommended  for 
large-size  motors. 

When  a  synchronous  motor  is  started  by  a  direct  current  or  by  an 
auxiliary  motor,  a  lamp  of  higher  voltage  than  that  of  the  supply-source 
may  be  connected  in  series  with  the  motor.  Owing  to  its  high  resistance, 
this  lamp  burns  feebly  at  first  under  the  action  of  the  E.M.F.  of  the 
source  of  supply.  In  proportion  as  the  motor  increases  in  speed,  if 
the  field -winding  is  connected  \vith  a  D.C.  source,  the  E.M.F.  which 
acts  on  the  lamp,  being  the  resultant  of  the  generator  and  motor  E.M.F.'s, 
will  produce  beats  or  pulsations  which  will  gradually  become  less  fre- 
quent. The  proper  time  to  connect  the  armature  directly  to  the  cir- 
cuit is  when  the  beats  have  become  extremely  slow,  and  when  the  lamp 
goes  out,  thereby  indicating  that  the  E.M.F.'s  are  in  opposition. 

When  the  motor  is  started  without  field -excitation,  as  an  induction- 


OPERATION  OF  SYNCHRONOUS  MOTORS.     HUNTING     109 


motor,  this  method  is  no  longer  applicable  and  the  phase-lamp  (or  else 
several  lamps  in  series)  must  be  connected  to  the  terminals  of  the  field- 
circuit,  which  is  then  subjected  to  the  inductive  action  of  the  variable 
flux  of  the  armature.     The  lamp  burns  brightly  at  first,  then  its  bright 
ness  diminishes  gradually,  and  the  extinction  of  the  lamp  shows  that 
synchronism   is   approaching;    the  direction  which   the   E.M.F.   will 
have  when  the  circuit  is  closed  is  not 
determined,   but    this  is  generally    of 
very  slight  importance. 

With  a  well-constructed  synchron- 
ous motor  there  is  no  need  of  a  phase- 
indicator,  because,  when  once  it  has 
attained  its  highest  speed  as  an  induc- 
tion-motor (which  may  sometimes  re- 
quire several  minutes),  the  powerful 
reactions  which  occur  when  the  ex- 
citation-circuit is  closed  are  sufficient 
to  insure  the  machine  falling  into  step. 

Current-controllers.  It  is  necessary, 
in  order  to  reduce  the  amount  of  current 
taken  from  the  line  during  the  starting 
period,  to  employ  a  current-controller, 
owing  to  the  absence  of  counter  E.M.F. 
in  the  armature.  Use  may  be  made 
for  this  purpose  of  a  simple  rheostat 
or  a  choke-coil.  Rheostats  are  more 
frequently  used,  on  account  of  their 
simplicity.  Switching  appliances  are 
used  for  the  purpose  of  connecting  them 
into  circuit  and  then  short-circuiting 
them,  an  armature  circuit-breaker 
being  interposed  to  break  the  circuit  in 
case  the  field-excitation  fails. 

Fig.  51  shows,  for  example,  the 
diagram  of  the  starting  apparatus 

of  a  certain  type  of  three-phase  motors  ranging  from  i  to  22  H.P.  When 
the  motor  is  at  rest,  the  pulley  is  on  the  idler  A,  the  triple-switch  C 
and  the  excitation  circuit-breaker  D  are  open. 

To  start  the  motor  the  triple-pole  switch  C  is  closed  on  the  first 
steps,  and  the  motor  then  receives  a  current  from  the  supply-trans- 


FIG.  51. 


110 


SYNCHRONOUS  MOTORS 


former  through  a  rheostat  which  limits  the  amount  of  current.  The 
motor  starts  and  attains  synchronous  speed  in  from  20  to  30  seconds. 
The  field-excitation  circuit  is  then  closed,  and  the  counterweight  of 
the  circuit-breaker  is  lifted,  which  enables  the  triple-pole  switch  to  be 
brought  to  the  second  position.  The  motor  then  receives  the  current 


At  Rest: 

The  belt  is  on  the  loose  pul- 
ley A.  The  circuit-breaker 
/  is  open.  The  switch  C  is 
to  the  left. 

Starting: 

1.  Throw  switch   C  to  the 
right. 

2.  Close  circuit-breaker  /. 

3.  When     synchronism     is 
reached  throw    switch  C 
to- the  left. 

4.  Shift  belt  from  pulley  A 
to  pulley  B. 

Stopping: 

1.  Shift  belt  to  pulley  A. 

2.  Open  circuit-breaker  7. 

N.  B. 

To  change  direction  of 
rotation  transpose  wires  a 
and  b,  andcandd.  (Fig.  51.) 


FIG.  52. 

directly,  and  it  attains  full  speed.  The  belt  can  then  be  transferred 
gradually  from  the  idler  pulley  to  the  driving  pulley  B. 

To  stop  the  motor  we  proceed  in  inverse  order.  The  belt  is  shifted 
to  the  idler  pulley  A,  the  counterweight  is.  dropped  (which  causes  the 
three-pole  switch  to  open),  and  finally,  the  circuit-breaker  D  of  the 
field-excitation  circuit  is  opened. 

For  motors  of  25  to  50  H.P.  the  single  resistance  is  replaced  by  a 
variable  rheostat  and  the  switches  have  but  one  step.  Once  the  speed  is 


OPERATION  OF  SYNCHRONOUS  MOTORS.     HUNTING     111 


attained   the    rheostats  are   short-circuited.     For  these  large   motors, 
separate  exciters  are  generally  used,  as  already  seen  (Fig.  52). 

The  operation  of  starting  without  load  does  not  require  more  than 
20  to  30  seconds,  and  it  is  effected  with  a  current  not  exceeding  the  nor- 
mal full  load  current. 


FIG.  53. 

It  will  be  observed  that  the  process  of  starting  involves  the  use  of 
high-tension  switches  on  the  primary  circuit,  to  isolate  the  transformer; 
also  fuses  on  both  the  primary  and  secondary  circuit. 

The  belt  is  shifted  from  the  idler  to  the  driving  pulley  in  the  ordinary 
way  by  a  belt-shifter.     Fig.  53  shows  the  manner  in  which  this  shifter 
should  be  placed  to  correspond  properly  with  the  direction  of  rotation 
and    the    direction    of    driving,    for    an 
observer  placed  at  the  commutator  end. 
Figures  A  and  B  refer  to  the  case  where 
the  direction  of  rotation    (indicated   by 
the  arrows)  is  the  same  as  the  hands  of 
a  watch ;  and  Figs.  C  and  D  refer  to  the 
case  where  the  rotation  is  in  the  opposite 
direction. 

The  use  of  a  reactance-coil  instead  of 
a  rheostat  has  the  objection  of  increas- 
ing the  lag  of  the  current  taken  from  the 
supply-source,  which  current  already  has 
too  much  lag.  It  is  therefore  preferable 
to  replace  it  by  an  auto -transformer 
which  enables  the  current  of  the  motor 
to  be  increased  at  low  voltage  while 

reducing  the  amount  of  current  taken  from  the  line.  Fig.  54  shows 
an  example  of  this  kind  of  arrangement  which  is  in  extensive  use. 
The  diagram  represents  the  simplest  case,  i.e.,  that  of  a  single -phase 
alternating  current  machine.  It  is  a  transformer  with  a  single  wind- 


FIG.  54. 


112  SYNCHRONOUS  HONORS 

ing  having  taps  connected  to  contacts,  over  which  moves  the  switch  b. 
The  two  ends,  A,  B,  of  the  winding  are  connected  to  the  source  of 
current-supply,  while  the  motor  is  connected  to  the  end  B  and  the 
switch  b. 

The  winding  operates  like  a  potentiometer  by  which  any  fraction 
whatever  of  voltage  can  be  obtained  at  will.  At  the  beginning,  very 
low  voltage  is  obtained  by  placing  the  switch  handle  to  the  right-hand 
side.  The  handle  is  then  moved  gradually  toward  the  left  in  pro- 
portion as  the  speed  increases,  and  gradually  raising  the  voltage  applied 
to  the  motor,  while  reducing  the  current  consumed  by  it.  The 
current  I\  taken  from  the  supply-source  is  a  fraction  of  the  current 
I2,  consumed  by  the  motor,  which  is  represented  by  the  inverse  ratio 
of  voltages 


The  winding  of  the  transformer  is  traversed  by  the  current  /i, 
between  A  and  D,  and  by  the  current  72—  /i,  between  D  and  B. 

In  the  case  of  polyphase  apparatus  it  is  sufficient  to  install  as  many 
transformers  with  commutating  switches  as  there  are  phases.  All 
the  levers  of  these  switches  are  operated  by  one  handle.  An  arrange- 
ment of  this  kind  was  utilized  with  polyphase  current  in  the  power- 
transmission  installed  for  Messrs.  Menier,  at  Noisiel  in  1893  (Power- 
Transmission  to  Noisiel,  Lumiere  Electrique,  May,  20,  1894). 

Starting  of  Single-Phase  Machines.  These  machines  can  also  be 
started  by  the  first  two  methods,  i.e.,  by  means  of  an  auxiliary  D.C. 
motor,  or  by  a  small  induction-motor  which  starts  by  means  of  an 
auxiliary  phase  such  as,  for  example,  "  shaded  pole  "  single-phase 
motors.  It  is  desirable  to  be  able  to  increase  the  speed  of  the  machine 
gradually  while  keeping  constant  the  speed  of  the  small  auxiliary  motor, 
because  the  latter  would  be  stalled  if  it  were  made  to  slow-up.  To 
overcome  this  difficulty  the  motor  is  mounted  on  guides,  and  it  transmits 
power  to  the  larger  machine  by  means  of  a  friction-cone  bearing 
against  a  large  disk  mounted  on  the  shaft  of  the  machine  itself.  The 
motor  is  gradually  shifted  on  the  guides  in  such  manner  that  the  fric- 
tion-cone approaches  the  center  of  the  disk  until  the  moment  when 
synchronism  is  attained. 

The  starting  of  the  machine  as  an  induction-motor  is  more  difficult, 
but  it  can  be  accomplished  by  transforming  temporarily  the  armature 
into  a  two-phase  armature,  by  means  of  a  second  winding  suitably 


OPERATION  OF  SYNCHRONOUS  MOTORS.     HUNTING     113 

provided  for  that  purpose,  and  which  can  be  of  small  wire,  inasmuch 
as  the  time  required  for  starting  is  so  short.  By  sending  into  this  aux- 
iliary winding  a  current  which  is  more  or  less  out  of  phase  with  respect 
to  that  in  the  principal  winding,  the  armature  produces  a  revolving 
field  which  is  more  or  less  imperfect  and  pulsating,  but  which  is,  never- 
theless, sufficient  to  start  the  motor. 

To  produce  the  phase-difference  between  the  two  currents,  recourse 
may  be  had  to  some  one  of  the  methods  knowrn  for  producing  induction- 
motors  by  means  of  a  single  alternating  force.  For  example,  use  may 
be  made  of  the  Tesla  method  of  introducing  a  self-induction  into  one 
branch  of  the  circuit,  and  a  resistance  in  the  other,  or  use  may  be  made 
of  the  Leblanc  method,  which  is  more  perfect,  and  which  consists  in 
introducing  a  self-induction  in  one  side  and  a  capacity  or  a  polarizer 
in  the  other. 

But  in  order  to  reduce  to  a  minimum  the  current  taken  from  the 
line,  there  is  again  a  great  advantage  in  inserting  a  reducing  trans- 
former, built  according  to  the  principle  already  explained,  and  supply- 
ing current  either  for  one  phase  or  for  both  phases  of  the  motor.  One 
arrangement  consists  in  supplying  only  the  principal  winding  by  means 
of  the  reducing  transformer  and  in  connecting  the  secondary  winding 
directly  to  the  switchboard  bus-bars.  In  such  a  case,  the  secondary 
winding  is  made  with  a  very  high  number  of  turns  so  as  to  have  a  very 
high  self-induction  and  to  absorb  only  a  weak  current  having  much 
lag  with  respect  to  the  current  of  the  other  winding. 

The  arrangement  employed  by  Labour  (Fig.  52)  is  the  reverse 
of  this.  In  this  case,  it  is  the  supplementary  winding  of  the  armature, 
made  of  fine  wire,  which  is  supplied  by  a  reducing  transformer,  while 
the  principal  winding  receives  current  direct  from  the  switchboard 
through  a  rheostat.  The  double-throw7  switch  C  being  moved  to  the 
right  (starting  position)  as  shown  in  the  diagram,  the  supplementary 
winding  will  then  be  in  circuit  with  the  secondary  of  the  reducing  trans- 
former, and  at  the  same  time  the  rheostat  is  inserted  in  the  principal 
circuit,  which  is  closed  by  the  circuit-breaker  7.  The  motor  starts  as 
an  induction-motor.  When  the  speed  approaches  synchronism,  the 
switch  C  is  suddenly  thrown  to  the  left;  by  this  operation  the 
current  in  the  supplementary  winding  is  broken,  and  the  rheostat  of 
the  principal  winding  is  short-circuited,  at  the  same  time  that  the 
fields  are  excited.  The  motor  is  then  in  step. 

In  large  motors  the  circuit-breaker/  is  replaced  by  a  starting  rheostat, 
the  object  being  to  avoid  fluctuations  in  the  lights  supplied  from  the  same 


114  SYNCHRONOUS  MOTORS 

distributing  system.  Sometimes  an  electromagnetic  circuit-breaker 
is  interposed  in  the  excitation-circuit,  its  function  being  to  open  the 
main-circuit  whenever  the  excitation -current  fails. 

The  operation  of  starting  and  synchronizing  requires  about  30 
seconds.  The  maximum  current  taken  from  the  mains  during  this 
period  does  not  exceed  the  normal  current  by  more  than  20  per  cent; 
and  it  falls  down  to  J  or  ^  the  normal  current  when  synchronism  is 
attained. 

Besides  these  general  methods,  there  is  still  another,  which  can  be 
used  only  in  the  case  of  motors  having  laminated  field-poles.  The 
method  consists  in  sending  alternating  currents  through  the  field- 
coils  and  through  the  armature-coils,  which  are  then  provided  with  a 
commutator.  The  latter  can  be  adapted  either  to  the  principal  wind- 
ing, or  to  a  supplementary  winding,  which  serves  only  for  starting  and 
for  excitation.  This  arrangement  has  been  used  by  the  Fort  Wayne 
Company  in  the  United  States  [Churchward,  Eclairage  Electrique, 
Vol.  XVII,  p.  41].  At  the  time  of  starting,  the  current  is  sent  through 
the  armature  and  field-circuits  connected  in  series;  the  machine  then 
operates  like  a  series-motor  having  a  commutator;  the  direction  of 
the  current  changes,  both  in  the  armature  and  in  the  field,  and  con- 
sequently the  torque  is  always  in  the  same 
direction  and  can  thus  cause  the  motor  to 
start. 

Both  the  field  and  armature  of  the 
Fort  Wayne  motor  are  laminated;  the 
field  has  two  windings,  one  being  a  fine 
wire  winding  for  synchronous  operation, 
the  other  being  a  coarse  wire  winding 
which  serves  for  starting.  The  armature 
itself  also  has  two  circuits  wound  on  a 
FIG.  55.  core  composed  of  sheet-iron  disks  of  the 

form  indicated  in  Fig.  55.  The  prin- 
cipal winding,  whicn  serves  for  synchronous  operation,  is  placed  in  the 
round  holes,  there  being  as  many  coils  as  there  are  field-poles,  and  the 
winding  is  connected  with  two  collector-rings.  The  other  winding, 
distributed  in  the  slots,  is  similar  to  that  of  an  ordinary  B.C.  drum 
winding  with  its  commutator. 

The  operation  of  starting  is  accomplished  by  means  of  a  single 
lever  controlling  the  switches.  If  it  is  first  placed  in  the  position 
indicated  in  Fig.  56  the  motor  will  be  connected  to  the  source  of  current 


OPERATION  OF  SYNCHRONOUS  MOTORS.     HUNTING     115 


like  a  series-motor,  and  it  will  start  as  such.  When  its  speed  attains 
synchronism,  a  centrifugal  device  closes  the  circuit  with  a  third  col- 
lector-ring on  the  right,  causing  an  incandescent  lamp  to  be  lighted; 
the  switch-lever  is  then  moved,  causing  the  normal  connections  to  be 
made;  the  armature  is  then  connected  directly  with  the  circuit;  and 
the  commutator  circuit  is  connected  with  the  fine  wire  excitation-cir- 
cuit. The  inductance  in  circuit  is  adjusted  in  such  a  manner  as  to 
permit  the  maximum  allowable  current  to  pass  through  the  circuit 
when  starting.  This  maximum  current 
is  about  two  to  two  and  a  half  times 
the  full  load-current. 

The  efficiency  of  this  type  of  motor 
is  70  per  cent  for  a  2\  H.P.  motor,  and 
80  to  85  per  cent  for  motors  of  10  to  15 
H.P.  They  start  readily  under  load, 
without  excessive  sparking  at  the  brushes, 
the  latter  being  of  carbon  and  having  a 
fixed  position.  The  power-factor,  which 
is  very  high  when  starting,  does  not  fall 
below  0.95  to  i  under  load. 

For  small  motors,  of  i  to  5  H.P.,  the 
Gramme  commutator  just  mentioned 
can  be  replaced  by  a  less  complicated 
commutator  having  only  one  segment 
per  pole  of  the  motor.  The  small  Ganz 

motors  are  of  this  kind.  Their  fields  are  excited  by  commutated 
currents  obtained  from  a  transformer  connected  in  shunt  to  the  line. 
The  even  segments  are  connected  to  one  terminal  of  the  field-circuit,  and 
the  odd  segments  to  the  other.  To  start  the  motor,  which  can  only  be 
done  without  load,  only  one  set  of  brushes  need  be  used  on  the  commuta- 
tor, the  others  being  put  off.  The  current  thus  changes  direction  in  the 
field-coils  and  in  the  armature  coils,  as  already  mentioned;  but  there  is 
much  sparking  at  the  brushes  during  the  whole  time  that  the  motor 
is  starting.  When  the  synchronous  speed  has  been  attained,  which 
requires  at  least  a  minute,  the  brushes  previously  removed  are  again 
put  on,  by  hand  or  else  by  a  centrifugal  regulator,  or  they  may  be  put 
in  circuit  by  means  of  a  switch ;  sparking  is  then  reduced  to  an  allowable 
amount.  Fig.  57  gives  the  diagram  of  the  arrangement  actually  used 
(Blathy  system).  The  field  and  armature  windings  are  in  parallel. 
The  armature-current,  coming  from  the  transformer  at  100  volts, 


FIG.  56. 


116 


SYNCHRONOUS  MOTORS 


through  the  wires  7,  77,  enters  into  the  armature  by  the  terminal  A, 
leaves  by  the  terminal  B  and  passes  through  the  secondary-winding 
•Si,  $2  of  the  transformer  E,  called  compensator,  which  plays  the  role 
of  self -inductance  for  throwing  the  armature-current  out  of  phase. 


FIG.  57. 

The  field-circuits  are  wound  for  a  low  voltage  (25  volts)  supplied  by 
the  secondary  circuit  Si,  $2  of  another  transformer  F. 

At  the  time  of  starting,  to  overcome  the  induction  of  the  field- 
circuit,  the  potential-difference  at  its  terminals  is  raised  by  the  primary 
winding  of  the  compensating  transformer  E,  whose  secondary  has  the 

armature-current  passing  through  it, 
as  already  stated.  The  field-current 
also  passes  through  a  fixed  resist- 
ance, ah,  in  the  rheostat,  R,  the 
object  of  which  is  to  reduce  the  lag 
in  the  field-circuit,  and  to  put  it  in 
phase  with  the  armature -current. 

The  motor  starts.  When  it  at- 
tains synchronism,  the  current  in  the 
field  is  commutated ;  the  connections 
are  then  changed  by  means  of  the 
double-throw  switch  D,  which  short- 
circuits  the  secondary  of  E,  connects 
the  two  neighboring  brushes  together 
and  sends  the  secondary  current  of 
FIG.  58.  F  into  the  variable  resistance  cd 

before  it  reaches  the  field-coils. 

A  motor  of  the  same  kind  was  put  on  the  market  some  years 
ago  by  the  firm  of  Brown-Boveri  (Fig.  58).  The  motor  starts  by  simply 
coupling  in  parallel  the  armature  and  the  field-circuit  and  a  resistance 


OPERATION  OF  SYNCHRONOUS  MOTORS.     HUNTING     117 

which  is  in  series  with  it.     To  bring  the  phases  of  the  two  currents 

into   proper  relation,   the  brushes  are   moved   an  angle   equal  to  — 

2 

during  the  operation  of  starting.  When  synchronism  is  attained,  the 
brushes  are  brought  back  to  their  place,  and  the  excitation-circuit 
is  connected  in  shunt  to  one  coil  alone  of  the  armature,  so  as  to  reduce 
the  voltage. 

Theory  of  Initial  Synchronizing.  Two  cases  must  be  considered, 
depending  on  whether  the  field,  when  separately  excited,  maintains 
its  normal  value  independently  of  the  speed,  or  whether  the  field  is 
produced  by  means  of  commutated  currents. 

Separate  Excitation.  Let  us  again  suppose  the  motor  to  be  con- 
nected to  a  constant  potential  source  E\  and  let  us  retain  the  same 
symbols  as  before.  Let  us  designate  by  T2  the  duration  of  the  period 
of  the  motor,  which  will  differ  from  T\  'the  period  of  the  supply-current, 
so  long  as  synchronism  is  not  attained.  For  simplicity,  let  us  write 

27T  27T 

TI  T2    W2' 

The  equation  of  the  current  proceeding  from  the  source  of  supply 
toward  the  motor  is 

•  ,  i  M     v     •  ^2  E,     .    , 

W+h~j:—Ei  sin  6^ E2  sin  (a)2t  —  6), 

dt  coi 

where  E2  equals  the  E.M.F.  induced  at  the  speed  of  synchronism, 
and  6  represents,  as  before,  the  lag  in  phase  of  the  motor.  From  this, 
neglecting  the  exponentials  which  occur  in  the  integral,  but  which 
always  neutralize  each  other  quickly,  and  taking 

(jt)\l  (t)2l 

tanco^ — ,     tan  <b=  — , 
r  r 

we  have 

E\  sin  (cuti  — (f)} 


Vr2+6J12/2  0)1  \ 

The  power  supplied  to  the  motor  at  each  instant  is 


rEisin 
-'}  L    v/r 


118  SYNCHRONOUS  MOTORS 

Integrating  between  t0    and  /,  the  work  done  is 


I     /"*£ 

W\  j    sin  (a)2t-0)  sin  (w2t-0  -<f>)dt. 

Jto 


While  the  second  term  increases  indefinitely,  the  first  is  doubly 
periodical  and  has  the  form 

sin  co\x  sin  ojvxdx. 


It  is  known  that  an  integral  of  this  form,  taken  within  limits  which 
comprise  a  whole  number  of  periods  of  w\x  and  of  co^x^  is  equal  to 
zero,  so  long  as  w\  is  different  from  aj2.  If  a)\  and  OJ2  are  very  dif- 
ferent from  each  other,  their  zeros  are  very  near  each  other  and  the 
amplitude  is  very  small.  On  the  contrary  when  w2  is  very  near  a>\ 
the  periodicity  of  the  integral  increases  in  length  to  the  point  of  becoming 
infinite. 

Two  cases  are  therefore  to  be  considered: 

i)  So  long  as  the  speed  of  rotation  is  low,  the  first  term  is  negligible 
in  comparison  with  the  second,  provided  that  the  inertia  in  the  motor 
is  sufficient  and  that  only  negative  work  is  done.  We  then  have 


T==--t^E2)   — ^      ^=cos</r 

2  \Wi         I     -  7  °     ' 

/  \  ' 

=  ~2(^£V 


2 

2    «//_ 


It  is  therefore  necessary  to  apply  mechanical  power  to  turn  the 
alternator  and  with  <£?2=the  speed  corresponding  to  the  period  72,  and 

,£?i  =  the  speed  corresponding  to  the  condition!  Q2=Qi  — ),  the  motor 

\  ^i/ 

torque  C^  will  have  the  following  expression: 

2 


,==--  ^£, 


E22 


OPERATION  OF  SYNCHRONOUS  MOTORS.     HUNTING     119 


Taking  x=—  we  will  have 


C,,=  - 


+  x2  tan2 


The  curves  in  Fig.  59   represent  Cw  as  a  function  of  x,  when  the 

E22 
constant     _      is  taken  as  unity,  and  when  tan  0  is  given  values  equal 

to  5  and  10. 


0.05 


It  is  seen  that  the  torque  Cw   increases  rapidly,  at  first,  with  the 
speed,  and  then  decreases  more  slowly  with  a  maximum 


X=—-' 

tan</>' 


To  bring  back  the  motor  to  a  speed  near  that  of  synchronism, 
it  would  be  necessary  to  exert  a  mechanical  force  (torque)  near  that 
which  corresponds  to  x=  i,  or 


C=- 


f'2 


28 


i.e.,  the  same  as  would  be  required  to  make  the  alternator  rotate  when 
short-circuited  upon  itself. 

2)  When  once  012  is  near  MI,  the  first  term  is  no  longer  negligible. 
The  integral  can  no  longer  be  evaluated  except  by  supposing  a>2  to  be 
constant  within  the  limits  of  integration,  i.e.,  during  a  great  number  of 
periods. 


120  SYNCHRONOUS  MOTORS 

Then,  taking  to  as  the  origin,  (/o=o),  we  will  have 


sn  wzt  —      sin  tot— 


cos 


=  ~-  f 

-—  f 

2  Jo 


4  0)\  —  0)2 

I          0)2 

—  sin 

Noting  that 

C02 


the  total  value  for  the  torque,  at  each  instant,  would  therefore  be 


When  ^i  is  very  near  &>2  the  amplitude  of  the  first  term  becomes 
very  high,  and  this  is  the  case  so  long  as  a>i  is  not  exactly  equal  to  &>2. 
At  that  moment,  there  is  a  discontinuity;  and  the  term  in  question 
vanishes. 

Under  these  conditions,  the  inertia  is  low  compared  with  the  enormous 
torque  which  is  developed,  and  the  motion  of  the  alternator  can  undergo 
pulsating  variations  of  speed  which  are  considerable.  The  hypothesis 
which  forms  the  basis  of  our  reasoning  concerning  the  nature  of  the 
motion  is  no  longer  fulfilled  and  the  equations  no  longer  have  any  value. 

Only  one  conclusion,  therefore,  can  be  drawn,  i.e.,  it  is  necessary 
to  bring  the  motor,  by  some  process,  to  a  speed  at  least  equal  to  that 
of  synchronism,  before  connecting  it  with  the  source  of  current-supply. 

A  second  method  of  starting  a  motor  without  load,  and  which  can 
be  used  with  success  every  time  there  is  only  one  power-transmission, 
consists  in  starting  simultaneously  the  generator  and  the  motor.  A 
very  weak  impulse  then  suffices  to  make  a>i=aj2,  at  the  moment  of 


OPERATION  OF  SYNCHRONOUS  MOTORS.     HUNTING     121 

starting.  The  two  machines  therefore  fall  readily  into  step,  and  if 
the  resisting  couple  due  to  passive  mechanical  resistances  (air-friction, 
bearing-friction,  hysteresis,  etc.)  is  not  too  high,  the  generator  drags 
the  motor,  in  its  progressive  acceleration,  the  two  always  remaining 
in  synchronism. 

Field  Due  to  a  Commutated  Current.  In  this  case,  the  effect 
produced  is  altogether  different.  Two  periods  must  again  be  dis- 
tinguished, i.e.,  whether  cu2  is  very  different  from  a>i,  or  whether,  on  the 
contrary,  it  is  of  the  same  order  of  magnitude. 

In  the  first  case,  i.e.  at  the  beginning  of  the  operation  of  starting,  the 
current  is  reversed  a  great  number  of  times,  both  in  the  armature  and 
in  the  fields,  while  the  armature  is  moving  an  angular  distance  corre- 
sponding to  a  period  T2.  The  motor  behaves  like  a  commutator- 
motor,  and  produces  a  torque  which  is  always  in  the  same  direction 
and  which  gradually  increases  the  speed. 

In  the  neighborhood  of  the  normal  speed,  the  effect  of  the  commutator 
on  the  excitation  becomes  apparent.  The  alternations  of  flux  are 
commutated  in  a  dissymmetrical  manner,  with  the  consequence 
that  a  very  powerful  field  with  slow  oscillations  is  produced,  on  which 
pulsatory  undulations  that  are  more  feeble  are  superposed.  This 
effect  may  be  expressed  by  the  formula, 


in  which  /o  has  a  value  which  is  variable  as  a  function  of  the  time. 

As  soon  as  the  two  periods  T\  and  T2  are  near  each  other,  the  small 
undulations  disappear  before  the  large  undulation,  namely,  the  undu- 

lation of  /o.     The  frequency,  —  ,  of  this  great  oscillation  is  evidently 

equal  to  the  least  common  multiple  of  —  and  —  .     The  period,  7"', 

1\  ±2 

therefore,  lengthens  indefinitely,  when  T2  approaches  7"i,at  the  same 
time  that  the  value  of  the  field  thus  produced  increases. 

Under  these  conditions,  the  power  furnished  by  the  motor  varies 
according  to  a  complicated  function.  The  oscillations  of  the  field 
can  be  taken  into  account  on  replacing  wto  by  aj't  and  substituting  the 
value 


-— =  -o0  sn  6/- 


122  SYNCHRONOUS  MOTORS 

in  the  expression  e2i.  But  the  equations  are  then  no  longer  rigorous  and 
can  only  be  used  to  predict  the  occurrence,  in  the  neighborhood  of  syn- 
chronous speed,  of  heavy  fluctuations  resulting  in  alternate  accelera- 
tions and  retardations  which  are  very  marked  and  for  which  the  inertia 
of  the  system  cannot  compensate.  As  soon  as  one  of  these  accelera- 
tions becomes  strong  enough,  it  will,  at  one  bound,  bring  the  speed  to 
its  normal  value,  leaving  the  motor  synchronized. 

The  starting  of  motors  of  the  Ganz  type  may  thus  be  explained 
approximately,  without  our  being  able  to  take  exactly  into  account 
the  effects  of  inertia.  It  is  known,  moreover,  that  synchronous  motors 
do  not  always  run  at  synchronous  speed,  but  at  a  speed  near  it,  which 
is  periodically  variable,  owing  to  the  production  of  the  alternating 
field  already  mentioned.  It  is  an  extreme  case  of  oscillations  of  load, 
which  we  now  proceed  to  discuss. 

Oscillations  of  Synchronous  Motors.  It  is  easily  observable,  by 
means  of  a  dead-beat  ampere-meter  connected  in  the  armature-circuit, 
that  the  operation  of  a  synchronous  motor  is  always  accompanied  by 
fluctuations  above  and  below  the  normal  current,  which  are  due  to 
oscillations  above  and  below  the  synchronous  speed.  With  a  good 
motor,  operating  in  a  satisfactory  manner,  these  oscillations  are  rapid 
and  slight,  except  when  the  load  undergoes  sudden  variations.  But, 
with  certain  motors,  these  oscillations  attain  a  considerable  amplitude 
and  they  are  complicated  by  slow  variations  of  very  high  amplitude, 
giving  rise  to  exaggerated  current-values  like  those  due  to  a  short- 
circuit,  which  usually  end  by  the  motor  falling  out  of  step;  it  may  even 
be  that  synchronous  operation  is  altogether  impossible. 

There  are  here  two  distinct  phenomena,  which  will  be  examined 
separately,  i.e.,  short-period  oscillations  of  the  motor  itself  and  long- 
period  parasite  oscillations  due  generally  to  foreign  causes. 

Short-Period  Oscillations.  Any  variation  of  the  speed  of  the  gen- 
erator or  any  sudden  variation  of  the  resisting  effort  at  the  motor-shaft 
gives  rise  to  an  oscillating  condition  of  motor-speed.  If  the  generator, 
for  example,  has  taken  the  lead,  the  driving  torque  becomes  stronger 
than  the  resisting  torque,  and  it  causes  the  armature  to  accelerate 
gradually;  but,  owing  to  inertia,  the  acceleration  lasts  longer  than 
would  be  necessary  to  attain  the  corresponding  phase  at  the  condition 
of  equilibrium,  and  the  motor,  in  its  turn,  takes  the  lead  in  phase. 
From  that  moment,  as  its  driving  torque  diminishes,  the  excess  of  the 
resisting  torque  tends  to  produce  a  retardation;  the  motor  slackens 
in  speed,  and  its  phase  is  soon  behind  what  it  should  be ;  and  so  on. 


OPERATION  OF  SYNCHRONOUS  MOTORS.     HUNTING     123 

Consequently,  oscillations  of  speed  occur  around  the  normal  speed 
value,  the  effects  being  altogether  analogous  to  the  oscillations  occur- 
ring in  the  speed  of  steam  engines  which  are  provided  with  imperfect 
governors. 

The  characteristic  feature  of  the  short-period  oscillations  of  the 
motor-speed  is  that  this  oscillatory  movement  occurs  without  loss  of 
synchronism,  in  other  words,  each  acceleration  in  one  direction  or  the 
other  is  stopped  by  an  opposing  torque  which  limits  the  amplitude, 
as  is  the  case  with  the  oscillations  of  a  pendulum. 

This  torque  is  easily  determined  by  the  variation  of  power  of  the 
two  machines  (A.  Blondel,  "Coupling  of  Alternators,'"  La  Lumiere 
Eletrique,  Vol.  XLV,  p.  352).  Let  us  first  suppose  these  two  machines 
to  be  similar;  let  PI  and  P2  represent,  respectively,  the  generator- 
power  and  the  motor-power.  The  variation  in  the  electric  power 
received  by  the  motor  and  transmitted  by  the  generator,  per  unit  of 
angle  a  of  angular  variation,  taking  2P  to  equal  the  number  of  poles, 
is,  according  to  the  formulae  on  page  38,  as  follows  : 


da      da        *\dO       dO 


—         (sin  f  cos  6). 

£ 

In  this  expression  Z  equals  the  total  impedance,  and  y  is,  as  before, 
the  angle  of  lag  between  the  current  and  the  resultant  E.M.F.,  the 
relations  being  such  that 

,tanr=^.       .. 

The  correcting  torque  is  obtained  by  dividing  this  power  by  the  mean 
angular  velocity 


From  this  we  have 


p2  \2p^  Eo  sin  Y  cos  0 

-         — 


] 


124  SYNCHRONOUS  MOTORS 

If  we  let  K  equal  the  inertia  of  the  motor,  the  period  of  oscillation 
produced  by  this  torque  may  be  deduced  by  the  known  formula, 


2E2  . 
whence,  on  substituting,  and  noting  that is  nothing  more  than  the 

Ct 

short-circuit  current,  Iac,  of  one  of  the  machines — the  motor — closed  on 
itself,  there  remains 

27T.   /  KtiJ 


p    ^Ai/.vcSin  7- cos  0 

or,  if  we  suppose  6  to  be  small  enough  so  that  cos  6  may  be  taken  as 
equal  to  unity,  we  will  have 


Kco 


In  the  case  where  the  machines  are  not  similar  and  are  running 
at  different  speeds  a\  and  #2,  it  can  be  shown,  in  the  same  way  (.4. 
Blondel,  loc.  cit.,  p.  357),  that  the  equation  of  oscillatory  movement 
is 


KA  (K2\ 

/  W) 


In  the  case  of  small  oscillations,  if  we  replace  the  numerators  by  their 
values,  we  have 

e<* 


and  assuming  cos  0=i,  and  sin  6  negligible  in  comparison  with  cos  6, 
we  have 


whence 


sin 


OPERATION  OF  SYNCHRONOUS  MOTORS.     HUNTING     125 
In  the  particular  case  where  the  motor  is  supplied  from  a  circuit  of 
great  output-capacity  the  term  ^-  can  be  considered  as  very  small  in  com- 
parison with  ^-,  and,  consequently,  the  expression  for  the  period  of 
K2 

the  oscillations,  as  a  function  of  the  constants  of  the  machine  and  of 
the  inertia  of  the  armature  and  of  the  parts  mounted  on  the  shaft,  is 
materially  increased. 

This  formula  shows  that  the  oscillations  are  all  the  more  rapid 
and  of  lower  amplitude  the  lower  the  self-induction  and  the  mechanical 
inertia. 

Consequently,  from  this  point  of  view,  it  would  be  more  profitable 
to  select  motors  with  a  low  armature-reactance,  a  conclusion  which  we 
had  already  reached  by  another  process.  The  oscillations  not  only 
have  the  effect  of  causing  the  motor  to  fall  out  of  step,  and  of  thus 
limiting  its  maximum  output,  but  also  of  producing  oscillations  of 
E.M.F.  and  current,  including  change  of  phase  (lag).  These  effects 
can  manifest  themselves  in  a  very  objectionable  manner,  on  the  circuit 
and  on  the  generator,  by  producing  fluctuations  of  voltage  which  are 
often  important,  especially  when  the  reactance  of  the  circuit  is  large 
and  the  circuit  is  thereby  made  very  sensitive  to  variations  in  wattless 
current. 

A  synchronous  motor  which  is  oscillating  can  therefore  cause  trouble 
to  its  neighbors  on  the  circuit,  and  it  may  even  render  lighting  from 
that  circuit  impossible.  For  this  reason,  great  care  should  be  taken 
to  regulate  perfectly  the  speed  and  phase  of  the  motor  by  means  of  a 
phase-indicator,  before  connecting  it  to  the  supply-circuit. 

Every  sudden  change  in  load  gives  rise  to  a  strong  oscillation  in 
the  phase-angle,  causing  E2  to  go  beyond  the  position  of  equilibrium 
corresponding  to  the  new  value  of  the  power.  If  this  oscillation  comes 
from  a  reduction  in  load,  it  can  cause  no  inconvenience;  if,  on  the 
contrary,  it  is  caused  by  an  excess  of  load,  the  phase-variation  may, 
in  the  first  impulse,  exceed  the  angle  of  stability,  and  the  machine  will 
fall  out  of  synchronism. 

Calculation  shows  (A.  Blondel,  loc.  cit.,  Lumiere  Electrique, 
Vol.  XLV,  p.  24)  that  the  relative  increase  of  load  A  which  can  be 
made  suddenly,  starting  from  an  initial  power  (—P2)eo  does  not  depend 
on  inertia,  but  only  on  the  two  ratios 

and          cos=*; 


126 


SYNCHRONOUS  MOTORS 


and  the  curves  in  Fig.  60  enable  its  value  in  each  particular  case  to 
be  calculated  as  a  function  of  the  data.  These  curves  show  that  J 
varies  in  the  inverse  ratio  of  y  and  in  direct  ratio  of  x,  so  long  as  2  is 
lower  than  a  value  near  J.  The  conditions  that  insure  the  best  stability 
in  the  case  of  sudden  overloads  are,  therefore,  substantially  the  same 
as  those  which  insure  the  maximum  theoretical  output.  In  particular, 
in  the  case  where  E2=Ei,  we  should  have  cos  <^>=J,  whence  ml=r\/^. 
The  experiments  made  on  a  Ganz  motor  (Experiments  made  by 
the  Frankfurt  technical  commission,  Upperborn,  Lumiere  Electrique, 
Vol.  XXXVI,  p.  315),  showed  that,  starting  from  no-load,  a  sudden 

load  equal  to  1 50  per  cent  of  the  nominal 
load  may  be  put  on  the  motor.  This  load 
could  be  much  exceeded  for  alternators 
of  lower  inductance,  especially  if  they  are 
provided  with  dampers  on  the  pole- 
pieces. 

Damping  of  Oscillations.  The  surest 
way  to  prevent  oscillations  of  both 
kinds  from  attaining  an  excessive  ampli- 
tude consists  in  damping  them  rapidly. 
It  is  an  application  of  the  general 
principle  laid  down  in  a  masterly  man- 
ner by  Cornu  ("  On  the  synchroniza- 
tion of  oscillating  systems,"  Journal 
de  Physique  et  Comptes  Rendu  de 
V Academic  des  Sciences,  31  Mai,  1887; 
Bulletin  de  la  Societe  des  Electriciens), 

who  showed  that  perfect  synchronization  can  be  obtained  only  when 
there  is  damping,  i.e.,  when  the  speed-variations  cause  a  supplementary 
expenditure  of  energy.  This  supplementary  expenditure  of  energy 
can  only  occur  to  a  slight  extent  in  the  armature-circuit,  but  it  is  pos- 
sible to  bring  it  about  in  the  field-circuits  by  allowing  eddy  cur- 
rents to  be  produced  in  the  massive  pole-pieces  or,  better  still,  in  special 
circuits  wound  on  the  fields,  and  short-circuited.  These  circuits 
constitute  the  damper  of  Hutin  and  Leblanc.  (C.  F.  Guilbert,  Lumiere 
Electrique,  Vol.  XL VI,  1892,  p.  801;  and  Leblanc,  Bull,  de  la  Soc. 
Int.  des  Electriciens,  1898).  This  damper  consists  either  of  bars 
of  copper  surrounding  the  pole -pieces,  as  shown  in  Fig.  61,  or,  better 
still,  of  a  series  of  bars  passing  through  the  fields  and  united  on  each 
side  by  copper  rings,  as  shown  in  Fig.  62. 


OPERATION  OF  SYNCHRONOUS  MOTORS.     HUNTING     127 

The  explanation  of  the  role  of  these  dampers  is  made  easy  by 
reference  to  the  elementary  considerations  set  forth  on  page  1 1 .  When 
there  are  no  speed-oscillations,  the  armature-reaction  flux  undergoes 
no  displacement  with  respect  to  the  field-poles,  and,  consequently,  it 
does  not  generate  any  induced  E.M.F.  in  the  damping  circuits.  On 


B 


FIG.  61. 

the  contrary,  any  variation  of  speed  producing  a  flux-oscillation 
gives  rise  to  eddy  currents,  which  are  all  the  stronger  the  greater  the 
conductivity  of  the  damping  circuits  and  the  greater  the  extent  to  which 
they  increase  the  flux  developed  by  the  armature.  The  energy  con- 
sumed by  the  induced  currents  is  what  produces  the  damping.  It 


FIG.  62. 

manifests  itself  by  the  production  of  a  corrective  torque  exerted  on  the 
armature  and  which  tends  to  bring  the  speed  to  synchronism. 

The  same  explanation  applies  in  the  case  of  a  single-phase  motor, 
with  this  difference,  that  the  inverse  revolving  flux  due  to  the  armature 
should,  in  consequence  of  its  high  velocity  relatively  to  the  damping 
circuit,  produce  strong  induced  currents  which  almost  completely 


128 


SYNCHRONOUS  MOTORS 


neutralize  this  armature-flux.  The  result  is  that  the  armature-reaction, 
or  the  apparent  inductance  in  the  armature,  is  reduced  about  one-half 
and,  at  the  same  time,  the  noise  of  single-phase  alternators  disappears. 
The  loss  of  energy  is  much  lower  than  might  be  supposed,  because 
it  eliminates,  incidentally,  the  hysteresis-effect  in  the  fields.  In  any 
case,  the  damping  effect  can  be  regulated  easily  to  an  amount  that  is 
deemed  sufficient,  as  can  be  seen  in  the  following  manner: 

Suppose  the  motor  departs  from  synchronism;  the  period  of 
its  induced  E.M.F.  is  no  longer  the  value  T,  but  a  slightly  different 
value,  Tf.  The  coefficient  of  mutual  induction  between  the  armature- 
circuit  and  the  damping-circuit,  being  a  periodic  function  of  the  angle 
described  by  the  moving  part,  can  be  written  in  the  following  form: 


M=M0sm 


^  -A 


in  which  7-  is  the  phase-angle  measured  from  the  origin  of  time. 
The  current  can  be  represented  by  the  following  expression: 

7= /o  sin  27i—  +  V  sin  (27^-, 

The  flux  sent  into  the  damping-circuit  by  the  induced  current,  at 
each  instant  will  be 


=  MI=  MQ!Q  Sin  271—  Sin   I  27T^7  -  f  \ 


--      sn 


or,  substantially,  if  we  confine  ourselves  to  the  first  term, 


+  cos    ~(~£)  +T\ 
The  corresponding  E.M.F.  will  be 


(i) 


-  /  sin    «- 


OPERATION  OF  SYNCHRONOUS  MOTORS.     HUNTING     129 

Consequently,  if  p  equals  the  resistance,  and  X  the  inductance  of 
the  damping  circuit,  the  induced  current  i  is  due  to  the  superposition 
of  two  currents  of  two  different  frequencies: 


The  energy  consumed  is  represented  by  the  integral 

W=  Ceidt, 

Jo 

which  contains  products  of  terms  having  different  frequencies.  If  the 
integration  is  supposed  to  extend  over  a  sufficiently  long  period,  these 
terms  can  be  neglected  in  comparison  with  the  others,  which  alone 
increase  indefinitely.  In  that  case,  the  value  of  the  mean  power 
expended  per  second  can  be  written  as  follows: 


The  first  term  is  substantially  constant,  since  T'  remains  near  T. 
On  the  contrary,  the  second  term,  which  vanishes  when  T'=  T,  increases 
rapidly  with  the  difference  of  frequencies.  It  represents  the  correlative 
increase  of  energy  consumed,  which  produces  the  damping. 

Supposing  the  difference  —  —  •=•  to  be  very  small,  the  inductance 

may  be  neglected,  in  comparison  with  the  resistance,  in  the  second 
term.  In  the  first  term,  on  the  contrary,  the  inductance  is  always  very 
large,  in  comparison  with  the  resistance,  and,  consequently,  the  latter 


130  SYNCHRONOUS  MOTORS 

may   be   neglected.     The   expression   for   P   is   therefore   practically 
equal  to  the  following: 

/  -»  JT      Y    \     n  t~~  /  \  ~1  O 

.-•     (3) 


Consequently,  by  making  p  very  low,  as  was  done  by  Hutin  and 
Leblanc,  the  permanent  expenditure  of  energy  due  to  the  current  of 

frequency  j;  +  j^  can  be  reduced,  at  the  same  time  that  the  damping 

effect  due  to  the  current  of  frequency  — — -=?  is  amplified. 

When  the  pole-pieces  are  solid,  they  already,  by  themselves,  play 
the  role  of  a  damping  circuit,  and,  their  resistance  p  being  very  small, 
it  can  be  readily  understood  that  the  effect  thus  obtained  would  be 
already  quite  appreciable. 

At  synchronism,  we  have  T'=  7\,  and  the  preceding  expressions 
(i)  and  (2)  (p.  128)  reduce  to  the  form 


2     L  L     ,V+«)'-<H        •# 

o-rr  .  /  .         / 

and 


The  armature-reaction  then  produces  two  fluxes  in  the  field-coils, 
one  being  constant,  which  represents  the  mean  reaction,  and  the  other 
having  a  frequency  which  is  double  that  of  the  alternators. 

Long-Period  Oscillations.  Long-period  oscillations,  characterized 
by  periodical  variations  of  current  which  can  be  observed  on  an  ampere- 
meter, may  occur  in  consequence  of  causes  which  are  apart  from  the 
peculiarities  of  the  motors  themselves.  For  example,  a  motor  which 
drives  a  machine-tool  subjected  to  a  variable  load,  or  a  motor  connected 
to  a  generator  whose  speed  varies  periodically  (for  example,  a  generator 
driven  by  a  steam  engine  having  a  single  cylinder),  will  itself  undergo 
the  same  corresponding  oscillations  of  currents,  which  are  much  slower 
than  the  preceding.  These  slow  oscillations  have,  moreover,  the 
incidental  effect  of  keeping  up  the  real  oscillations,  and  thus  they  are 
often  the  cause  of  the  machines  falling  out  of  synchronism. 

Other  causes  more  complex,  and  which  are  still  less  well  understood, 
can  bring  about  the  systematic  and  periodic  production  of  this  phenom- 


OPERATION  OF  SYNCHRONOUS  MOTORS.     HUNTING     131 

enon  of  successive  falling  out  of  and  falling  into  synchronism.  For 
instance,  Hutin  and  Leblanc  (M.  Leblanc,  Lumiere  Electrique,  Vol. 
XXXIII,  p.  227,  and  F.  Geraldy,  ibid.,  Vol.  XLVI,  p.  652)  showed 
by  an  experiment  that  some  Ganz  motors  of  the  kind  excited  by 
commutated  currents  did  not  always  attain  synchronism,  but,  in  cer- 
tain cases  (especially  when  running  without  load),  assume  an  oscillatory 
movement  having  long  periods.  These  periods  can,  in  such  a  case, 
be  attributed  to  slow  alternations  of  the  excitation  produced  by  the 
current-commutator.  This  is  an  additional  reason  for  abandoning 
definitely  this  kind  of  excitation.  Certain  motors  also  appear  subject 
to  slow  oscillations,  in  consequence  of  the  discontinuity  of  the  V-curve 
produced  by  the  presence  of  important  upper  harmonics  in  their  E.M.F. 
curve,  as  already  noted  above  (page  103). 


CHAPTER  V 
TESTS  OF  SYNCHRONOUS  MOTORS 

Characteristic  Curves.  It  may  be  stated,  in  general,  that  the  char- 
acteristic curves  of  synchronous  motors  are  the  same  as  those  of  alter- 
nators, i.e.,  the  variation  of  the  inducing  flux  is  represented  as  a  func- 


880 
260 
240 
820 
800 
180 

10  160 
U 

cj  '40 

Q_ 

E  120 

< 

100 

80 
60 
40 
?0 
0 

^ 

S 

100 
90 
80 
70 
60 

(0 

50  -H 
40    O 

30 
2O 

10 

0 

S 

/ 

^ 

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/ 

/ 

/ 

/ 

f 

/ 

j 

f 

f 

^ 

f 

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/ 

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t 

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/ 

t 

/ 

1 

f 

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/ 

/ 

I 

f 

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-> 

Field  -  Exci^a  tion.  Amperes 

FIG.  63.— Current  consumed  by  a  15  H.P.  synchronous  motor,  run  at  constant 
potential  with  variable  field-excitation.  The  ordinates  represent  armature 
current  (amperes)  per  phase. 

tion  of  the  exciting  ampere-turns  by  an  "  excitation-curve,"  and  the 
armature-reaction  is  represented  by  a  short-circuit-curve  obtained 
according  to  the  method  of  Behn-Eschenburg,  by  measuring  the  amper- 
turns  necessary  to  produce  increasing  currents,  when  the  machine  is 

132 


TESTS  OF  SYNCHRONOUS  MOTORS 


133 


short-circuited  on  itself  and  is  run  at  a  speed  near  its  normal  speed. 
The  short-circuit  current  will  be  all  the  smaller,  for  a  given  excitation, 
the  greater  the  armature-reaction.  Since  a  good  stability  of  operation 
is  only  obtained,  as  already  seen,  with  low  armature-reaction,  it  follows 
that  a  good  synchronous  motor  will  be  characterized  by  a  short-cir- 
cuit curve  which  is  rapidly  ascending  such  as  that  shown  in  Fig.  63, 
which  refers  to  a  synchronous  motor  with  revolving  iron  of  Oerlikon 
type,  described  by  Kolben  (Elektrotechnische  Zeitsckrift,  19  Dec. 


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10 

0 

/                   2                  3                 4                  5 

Amperes 

FIG.  64. — Power  input  for  15  H.P.  synchronous  motor,  run  with  variable  field- 
excitation. 

1895).  A  motor  is  good,  according  to  this  author,  when  a  short-cir- 
cuit-current equal  to  the  normal  operating  current  is  obtained  by  an 
excitation  giving  not  more  than  one-third  the  normal  voltage,  for 
the  induced  E.M.F.  Finally,  from  what  precedes,  there  exists,  for 
synchronous  motors,  a  third  kind  of  important  characteristic  curve, 
the  V-curves,  of  which  the  theory  has  already  been  given.  These 
curves  are  drawn  for  variable  loads  applied  by  a  Prony-brake,  by  measur- 
ing, for  each  value  of  the  excitation,  the  current-values  given  by  an 
ampere-meter,  and  the  electric -power  values  given  by  a  watt-meter. 


134 


SYNCHRONOUS  MOTORS 


20 
18 

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«,B 

olO 
~  8 
*  6 
4 

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fa 

"**«< 

Y"«4 

' 

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l-i 

-* 

-> 

Kj- 

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k 

x< 

^ 

% 

K* 

-', 

OH 

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-N 

55 

f 

*^N 

h*< 

,/ 

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, 

'.V 

-* 

•* 

•*- 

*- 

•*- 

01         8345 
A  m  peres 

FIG.  65. — Power-curves  for  three- 
phase  30  H.P.  synchronous 
motor,  run  with  variable  field- 
excitation. 


Figs.  64  and  65  give,  for  example,  these  curves  obtained  by  Kolben 

for  an  Oerlikon  three-phase  synchronous  motor  of  15  H.P.     These 

curves  show  that  the  minimum  current 
in  the  armature  is  obtained  with  about 
the  same  excitation  at  all  loads,  and 
that  the  efficiency  is  almost  constant 
within  wide  limits.  For  example,  the 
power  consumed  at  full  load,  to  pro- 
duce 14  k.w.,  scarcely  varies  between 
i  and  5  amperes  of  excitation.  These 
favorable  conditions  characterize  a 
motor  having  low  armature-reaction, 
Motors  having  high  armature-reaction 
give  rise  to  eddy  currents  in  the 
pole -pieces,  which  involve  a  consider- 
able increase  in  the  power  absorbed, 
when  the  current  increases  in  the 

armature.      This    peculiarity   may  be    made    evident    by   means    of 

a  fourth  characteristic  curve  known  as  the  "  characteristic  curve  of 

loss   by   parasite-effect."      This 

curve  is  obtained   by  taking  as 

ordinates    the    power    absorbed 

when    running     short-circuited, 

the     excitation-currents     being 

taken  as  abscissae.     Figs.  66  and 

67  show,  quite  clearly,  the  differ- 
ence, in  this  respect,  between  a 

motor  having  a  high  armature- 
reaction  and  a  motor  having  a 

low  armature-reaction.      In  the 

latter       the       eddy      currents, 

which  assume  great  importance, 

impart  a  parabolic  form  to  this 

characteristic  curve. 

Measurement  of  Efficiency. 

The  most  natural  method  for  the 

measurement    of    efficiency     is 

the     Prony-brake-dynamometer 

method,  used  in  the  same  way  as  for  a  D.C.  motor.     The  resisting 

torque  produced  by  the  brake  is  measured,  and,  at  the  same  time,  the 


ISO 


L 


A, 

/ 

, 

* 

/ 

1 

- 

/ 

/ 

/ 

/ 

>r 

^ 

1 

/ 

1 

/ 

rx 

I 

/ 

# 

s 

/^ 

/ 

^' 

S 

J 

Y' 

/ 

X-' 

s 

£-- 

<*' 

i 


F~i.e/J-£:xe./-af(.os7 ,  / 

FIG.  66. — Curves  for  30  H.P.  synchronous 

motor  having  high  armature-reaction. 
A=  characteristic  curve  without  load; 
B  =  short  -circuit  current; 
C= power  absorbed  when  short-circuited. 


TESTS  OF  SYNCHRONOUS  MOTORS 


135 


electric  power  applied  at  the  terminals  of  the  motor  is  measured  by  a 
watt-meter.  This  method  has  the  objection  of  requiring  the  expendi- 
ture of  too  much  energy  in  the  case  of  large  motors. 

The  author's  method,  described  further  on,  for  the  study  of  con- 
ditions of  operation,  then  finds  its  application.  In  such  cases,  the 
generator  and  motor  are  mounted  and  coupled  together  in  such  manner 
that  the  generator  is  driven  by  its  pulley,  and  also  tends  to  be  driven 
by  the  motor  which  is  connected  to  the  other  end  of  the  shaft  and 
which  takes  the  lead  and  drags  the  dynamometer.  The  latter 


140    S60 


IZO    240 
JOO    ZOO 


60 


40      60 
ZO      4O 


6000 


ZOOO 


Field -Exct  fa f/orr  ,  Amperes 

FIG.  67. — Curves  of  15  H.P.  synchronous  motor.     Normal  voltage  240  volts. 
A  =  characteristic  curve  without  load. 
B  =  short-circuit  current. 

then  measures  directly  the  work  done  by  the  motor,  and  all  that  is 
necessary  is  to  measure,  by  means  of  a  watt-meter,  the  electric  power 
supplied  to  the  motor  by  the  generator.  If  we  let  C  equal  the  resisting 
torque,  n  the  number  of  turns  per  second,  and  w  the  electric  power 
measured  by  the  watt-meter,  the  efficiency,  77,  will  be  expressed  by 
the  following  equation: 


The  segregation  of  the  different  losses,  in  a  synchronous  motor,  is 
a  much  more  complex  problem  than  in  the  case  of  direct  current  machines 
because  the  parasite  effects  (hysteresis  and  eddy  currents)  in  the 
field-pole  pieces  depend  not  only  on  the  induced  current  and  on  the 


136 


SYNCHRONOUS  MOTORS 


excitation-current,  if  the  armature-core  is  slotted,  but  they  also  depend 
on  the  phase-angle  of  the  induced  current,  especially  in  the  case  of 
single-phase  motors.  It  can  be  readily  understood  that,  in  the  case 
of  single -phase  motors,  the  variation  in  the  reaction-flux  in  front  of  the 
pole  is  all  the  greater  the  more  rapidly  the  armature -current  varies  in 
the  position  of  maximum  mutual  induction.  For  the  same  current  in 
the  armature,  the  losses  will,  therefore,  be  greater  the  smaller  the  phase- 
angle  (lag)  of  the  current. 

By  the  aid  of  these  assumptions,  the  losses  can  always  be  approx- 
imately separated,  when  the  conditions  of  operation  are  precise,  such 
as  when  the  current-supply  has  a  constant  voltage  and  the  excitation- 
current  is  constant.  In  such  cases,  the  losses  by  friction  are  measured 


401 


-40.1 


100  # 

90 

80 

70 

60 

SO 

40 

30 

20 

IO 


10     20     50    4O    5O    6O    TO    SO    SO 
FIG.  68. 

by  running  the  machine  without  excitation,  by  means  of  a  small  D.C. 
motor  whose  efficiency  curve  has  been  obtained.  On  exciting  the 
machine,  the  loss  becomes  increased  by  an  amount  equal  to  the  hys- 
teresis and  eddy  currents,  in  the  armature  and  the  fields,  other  than 
those  due  to  armature-reaction.  They  may  be  distinguished,  as  usual, 
by  running  at  various  speeds.  The  motor  is  then  run  with  a  load 
applied  at  the  brake.  The  difference  between  the  losses  noted  and 
those  previously  measured  consists  of  two  portions:  .the  loss  due  to 
resistance -heating  in  the  armature,  which  can  be  easily  calculated 
from  the  resistance  and  the  current,  which  are  known ;  and  the  increase 
of  loss  due  to  armature-reaction,  which  is  obtained,  more  or  less 
approximately,  by  taking  the  difference. 


TESTS  OF  SYNCHRONOUS  MOTORS 


137 


Fig.  68  gives  an  idea  of  the  manner  in  which  the  power-factor 
and  the  efficiency  vary  as  a  function  of  the  load  in  a  synchronous 
motor. 

Experimental  Tests.  It  is  interesting  to  compare  the  results 
of  experiments  with  those  of  calculation.  The  methods  which  can  be 
used  for  that  purpose  will  now  be  indicated. 

It  should  be  understood  that  a  satisfactory  agreement  between 
theory  and  practice  can  only  be  hoped  for  when  working  with  machines 
having  E.M.F.'s  of  the  same  wave-form,  and  as  nearly  as  possible 
sinusoidal.  Owing  to  the  want  of  this  precaution,  certain  authors 
(Bedell  and  Ryan,  loc.  cit.)  have  found  discontinuities  in  operation 
which  are  inexplicable  by  the  ordinary  theory 


FIG.  69. 

The  factors  to  be  measured  experimentally  are  the  E.M.F.'s  induced 
in  the  generator  and  the  motor,  the  current  in  amperes,  and  the  phase- 
angles  between  these  various  quantities,  and  also  the  electrical  and 
mechanical  powers  applied  or  developed.  The  investigation  can  be 
more  or  less  complete  according  to  the  apparatus  available. 

The  author  published,  in  1893  ("Theory  of  the  Coupling  of  Alter- 
nators," Bulletin  de  la  Societe  des  Electriciens,  1893,  p.  9),  a  method, 
based  on  that  of  Hopkinson  for  D.C.  dynamos,  which  has  the  advantage 
of  requiring  only  a  small  amount  of  mechanical  power  (Fig.  69). 

The  two  machines,  generator  A2  and  motor  AI,  are  placed  end  to 
end,  and  the  two  shafts  are  joined  together  by  a  flange-coupling  or  a 
torsion  dynamometer  C  (of  type  such  as  used  by  Mascart,  Rateau, 
Bedell,  etc.)  so  arranged  that  they  can  be  set  in  various  positions  for 
changing  the  angle  of  phase  of  the  two  shafts  with  respect  to  each 
other.  This  phase-angle  is  known,  at  rest,  and  all  that  is  necessary  is 
to  add  to  it  the  angle  of  torsion  of  the  dynamometer  to  have  the  phase- 


138  SYNCHRONOUS  MOTORS 

angle  when  running.  One  of  the  alternators  is  driven  by  a  belt  run 
over  its  driving  pulley,  and  it  drives  the  other,  mechanically  through 
the  dynamometer,  as  well  as  electrically  through  the  circuit.  The 
alternator  which  is  belt-driven  acts  as  a  motor,  the  other  acting  as  a 
generator;  the  torsion-dynamometer  connected  between  the  two 
machines  measures  the  excess  of  power  consumed  by  the  generator 
over  and  above  the  energy  recovered.  The  power  applied  at  the 
pulley  represents  the  total  losses  of  the  two  machines  operating  as  a 
power-transmission  system.  From  these  data  the  efficiencv  of  trans- 
mission may  be  deduced. 

Suppose  that,  for  each  phase-angle,  the  excitation-current  and  the 
line-current  are  read,  by  means  of  an  electrodynamometer  E,  also 
that  the  voltage  at  the  terminals  is  read,  by  means  of  a  volt-meter, 
and  that  the  electric  power  is  read  by  means  of  a  watt-meter  inserted 
in  the  circuit,  with  its  shunt-coil  connected  to  the  terminals. 

By  comparing  this  "actual"  power  with  the  "apparent"  power, 
the  phase-angle  between  the  internal  E.M.F.'s  EI  and  E2,  is  obtained 
substantially.  The  magnitudes  of  these  E.M.F.'s  are  known  from  the 
excitation-curves,  which  are  drawn  once  for  all.  This  method  is 
particularly  simple  when  no  transmission-dynamometer  is  used,  when 
the  loads  are  perfectly  steady;  but  the  loads  are  not  then  obtained  in 
terms  of  the  mechanical  power  of  the  motor. 

When  it  is  impossible  to  couple  the  two  machines  rigidly,  or  when 
they  do  not  have  the  same  number  of  poles,  it  is  not  so  easy  to  determine 
the  phase-difference  or  lag  between  the  two  motors,  and  it  is  then  neces- 
sary to  proceed  in  one  of  the  two  following  ways: 

(1)  It  is  possible  to  mount  on  the  shaft  of  the  machines  some  small 
alternators  serving  as  "  tell-tales."     They  are  arranged  in  such  man- 
ner that  their  E.M.F.'s  synchronize  in  phase  with  the  E.M.F.'s  induced 
when  running  without  load.     A  triple  oscillograph  record  will  then 
give,  directly,  for  each  load,  the  phase-angles  between  these  E.M.F.'s 
and  the  current. 

(2)  When  the  machines  have  not  the  same  number  of  poles  the 
phase-angles  between  the  induced  E.M.F.'s  can  be  measured  mechan- 
ically by  means  of  the  ingenious  phase-indicator  of  Bedell  and  Moler. 
The  two  machines  are  placed  end  to  end,  with  their  shafts  almost 
touching.     The    phase-indicator    consists    of    two    metallic   disks    23 
centimeters  in  diameter  and  8  millimeters  thick,  each  fastened  to  one 
end  of  the  shaft  and  provided  with  curved  slits  symmetrically  disposed 
in  the  form  of  an  Archimedean  spiral.     (Fig.   70.)     The  disks  are 


TESTS  OF  SYNCHRONOUS  MOTORS 


139 


arranged  in  such  a  way  that  their  spirals  are  pointed  in  opposite  direc- 
tions. Owing  to  the  form  of  these  curves,  the  distance  from  their 
points  of  intersection  to  the  center  is  proportional  to  the  phase-angle 
between  the  two  disks,  or  the  lag  of  one  with  respect  to  the  other. 
The  point  of  intersection  of  two  slits  forms  a  small  opening  traversed 
by  rays  of  light  which,  in  turning,  make  a  circle  that  expands  or 
contracts  in  proportion  to  the  phase-angle.  The  radius  of  the  circle 
can  be  measured  on  a  fixed  scale  which  is  graduated  directly  in 
angles, 


FIG.  70. 

Bedell  and  Ryan  have  used  this  method  to  determine  the  phase- 
angle  between  E.M.F.'s,  by  bringing  this  phase-angle  to  zero  when  the 
poles  are  in  line.  The  currents  are  measured  by  means  of  an  electro- 
dynamometer.  The  E.M.F.'s  are  deduced  from  the  excitation- 
curves.  Their  vectors  are  then  drawn  with  an  angle  between  them 
equal  to  the  measured  phase-angle,  and,  from  these,  the  resultant 
vector  is  obtained,  in  magnitude  and  in  phase,  by  completing  the 
parallelogram.  The  current  is  known,  in  magnitude.  Its  phase, 
with  respect  to  the  resultant  E.M.F.,  is  deduced  from  the  reactance, 
which  is  known  by  means  of  the  expression 


tan  r= 


reactance 
resistance' 


140  SYNCHRONOUS  MOTORS 

or  it  can  be  deduced  from  an  oscillograph  diagram.  We  then  have  all 
that  is  necessary  to  draw  the  diagram.  The  results  are  sufficiently 
concordant  with  the  theory,  but  they  indicate  no  new  phenomenon, 
except  the  discontinuity  already  mentioned.  For  more  complete 
details  see  the  author's  original  paper. 

Advantages  and  Disadvantages  of  Synchronous  Motors.  The 
characteristic  features  of  synchronous  motors,  constituting  their  advan- 
tages and  disadvantages,  may  be  summarized  as  follows: 

(1)  Their  construction  is  as  simple  as  that  of  alternators,  and  it 
enables  them  to  be  connected  directly  to  high  voltage  circuits. 

(2)  Their  specific  output  and  their  efficiency  are  also  the  same, 
and  are  quite  satisfactory. 

.(3)  When  once  put  in  operation  they  have  perfectly  uniform  speed. 

(4)  Their  power-factor  is  as  good  as  may  be  desired  when  their 
E.M.F.  is  properly  selected.     By  using  them,  all  reactive  currents  can 
be  eliminated,  and  currents  can  even  be  made  to  lead  in  phase,  by 
suitable  regulation  of  the  excitation. 

(5)  It   is   difficult   to   start   these   motors,   especially   single-phase 
motors;    special  methods  and  apparatus  must  be  employed  and  also 
an  idler-pulley  or  clutch.     Even  when   starting  without  load,  which 
is  the  only  practical  way,  a  very  high  current  is  required. 

(6)  Speed-regulation  is  impossible. 

(7)  A  very  sudden  overload  causes  the  motor  to  fall  out  of  phase 
and  brings  about  a  short-circuit. 

(8)  Any  irregularity  of  operation  of  the  generator  causes  great 
oscillations  of  speed   ("  pumping  ")   which  make  the  voltage  of  the 
system  fluctuate.     If  the  excitation  is  not  well  regulated,  the  motor 
produces  a  more  or  less  important  reactive  current,  which  may  bring 
about  a  drop  of  voltage  in  the  system. 

(9)  If  the  E.M.F.  curve  of  the  motor  is  different  from  that  of  the 
supply-system,  harmonics  will  be  produced,  which  will  uselessly  increase 
the  line-current  and  compromise  the  stability  of  operation.     Stability 
of  operation  can  only  be  attained  by  the  addition  of  inductance  to  the 
circuit;  and  it  remains  weak,  at  best. 

(10)  A  synchronous  motor  reacts  more  or  less  on  the  rest  of  the 
system,  to  which  it  can  communicate  its  oscillations,  or  of  which  it 
can    distort    the    E.M.F.    curve.     The    construction    of    synchronous 
motors  should  therefore  be  carefully  studied,  not  only  with  reference 
to  their  operation  by  themselves,  alone,  but  with  reference  to  their 
effect  on  the  distributing  system  from  which  they  are  to  be  supplied. 


CHAPTER  VI 

OTHER    MOTORS    OPERATING    SYNCHRONOUSLY    WITHOUT    DIRECT 
CURRENT  EXCITATION 

It  is  possible  to  devise  other  motors  which,  once  brought  to  a  suitable 
speed,  can  continue  to  operate  synchronously,  without  excitation 
by  a  direct  current  or  by  a  commutated  pulsating  current.  These 
motors,  not  being  used  industrially,  constitute,  properly  speaking, 
theoretical  curiosities,  but  it  is  interesting,  in  order  to'  give  a  general 
idea  of  the  whole  subject,  to  indicate  their  principles. 

Mention  can  be  made  of  two  types  of  such  motors;  synchronous 
motors  without  any  excitation,  known  as  "  reaction-motors,"  and 
synchronous  motors  with  alternating-current  excitation,  or  alternating- 
field  motors. 

Reaction  Synchronous  Motors.  It  is  natural  to  suppose  that  the 
constant  armature-reaction-field  of  polyphase  motors  can  be  used 
as  an  inducing  field.  It  is  sufficient,  for  this  purpose,  that  the  armature 
should  have  a  large  number  of  windings  to  produce  a  large  number 
of  ampere-turns,  and  that  the  current  passing  through  it  should  be 
much  out  of  phase,  in  order  that  the  magnetizing  turns  may  coincide 
with  those- of  the  field-poles.  Moreover,  the  air-gaps  should,  obviously, 
be  very  small,  to  reduce  the  reluctance  of  the  inducing  circuit.  The 
self -excitation  of  the  latter  is  then  only  an  exaggeration  of  the  effect  of 
•the  lagging  reactive  current,  which  has  already  been  mentioned  several 
times.  The  operation  of  such  a  motor  is  therefore  quite  possible, 
even  at  light  loads;  but  the  inductance  of  the  armature  has  been 
increased  at  the  same  time  as  its  reaction  on  the  field;  and  the  con- 
sequence is  that  the  stability  and,  moreover,  the  power-factor  of  the 
current  used,  will  be  small,  owing  to  the  great  lag  of  this  current.  Such 
a  motor  would,  therefore,  be  bad  from  a  commercial  point  of  view. 

The  conditions  of  operation  can  be  easily  ascertained  by  making 
use  of  the  diagram  on  page  94,  in  which  the  E.M.F.  due  to  external 
excitation  is  supposed  to  be  zero. 

The  operation  of  motors  on  the  reaction-principle  is  not  limited 

141 


142  SYNCHRONOUS  MOTORS 

to  polyphase  motors,  but  it  can  also  be  accomplished,  although  with 
more  difficulty,  in  the  case  of  single-phase  motors.  The  phenomenon 
was  first  observed  as  early  as  1874,  by  Siemens  of  London,  who  found 
that  he  could  suppress  the  excitation  of  a  single-phase  motor  without 
stopping  it.  The  explanation  is  simple  if  we  refer  to  the  use  already 
made  of  the  theorem  of  Leblanc,  i.e.,  that  the  alternating  field  of  the 
armature-reaction  can  be  replaced  by  two  fields,  each  of  half  the  strength, 
rotating  in  opposite  directions,  one  of  which  remains  in  front  of  the 
field-poles  and  magnetizes  them,  the  other  being  a  useless  parasite 
current,  rotating  with  twice  the  speed  of  the  poles  and  producing  no 
effect  (except  eddy  currents). 

Another  explanation  can  also  be  given,  based  on  the  variation  of 
the  coefficient  of  self-induction  of  the  armature  at  each  period  during 
its  rotation.  In  consequence  of  this  variation  the  reaction-flux  li  is 
a  maximum  when  the  armature  coils  are  exactly  in  front  of  the  poles. 

This  position  is  therefore  that  which  the  armature  would  tend  to 
take  if  a  direct  current  were  passing  through  it.  Hence,  if  the  machine 
is  driven  by  hand,  it  will  be  sufficient  that  the  motion  may  continue, 
as  in  certain  toys,  to  break  or  to  weaken  the  current  each  time  that  the 
coils  move  away  from  a  pole,  or  that  it  may  close  the  circuit,  or 
strengthen  the  current,  each  time  that  they  approach  the  next  pole. 
Now  this  effect  is  precisely  produced  with  an  alternating  current 
(because  the  direction  of  the  current  is  of  no  importance  here)  as  soon 
as  the  maximum  of  the  periodic  current  is  made  to  lead  in  phase 
with  respect  to  the  maximum  of  /,  as  can  be  easily  seen.  The  sum 
of  the  effects  which  attract  the  armature  in  the  direction  of  motion 
during  a  period  is  then  greater  than  the  sum  of  the  contrary  effects. 

The  motion  therefore  continues,  and  it  is  easy  to  see  that  it  may 
be  steady,  because  every  acceleration  of  the  motion  tends  to  diminish 
the  phase-angle  between  i  and  /,  and,  consequently,  tends  to  diminish 
the  dissymmetrical  action,  while  retardation  increases  the  phase-angle, 
and,  consequently,  increases  the  useful  action  of  the  current. 

These  considerations  can  be  easily  verified  by  calculation  by  referring 
to  the  equation 

,di 

u=e—n—cjl—. 
at 

in  which  we  will  let  e=o,  to  take  into  account  the  fact  that  there  is 
no  excitation.  The  alternator  having  a  variable  self-induction 

l=l'+X  COS  26t>/, 


MOTORS  OPERATING  SYNCHRONOUSLY  WITHOUT  D.  C.  143 

and  supposing  the  current  passing  through  it  from  the  inside  to  the 
outside  to  be  sinusoidal,  and  to  be  out  of  phase  by  the  angle  ^  with 
respect  to  the  position  of  coincidence  between  the  poles,  we  will  have 

7=/o  sin  (cot  —  f}. 
We  will  therefore  have,  for  the  voltage  at  the  terminals, 


u—  —T!  sin   a>t  — 

/  cos     cos  <w/  + 


—aj\  1  1'  --  J/o  cos  ^  cos  <w/  +  (/'  +  —  j/o  sin  (psmwt   , 

the  harmonics  of   the  order  3<i>/  introduced  by  the  pulsation  of  /  being 
neglected. 

This  equation  is  represented  by  the  diagram  in  Fig.  71.     It  is  seen 


FIG.  71. 

that  the  voltage  U  at  the  terminals  is  the  geometrical  resultant  of  the 
vectors 


~r7,  -^'+  j,-wr~j      (perpendicular) 

In  this  diagram  the  lag  <p  of  the  current  is  denned  as  the  interval 
between  the  zero-value  of  the  current  and  the  position  of  coincidence 
of  the  poles. 

It  is  seen  that  the  E.M.F.  of  self-inductance  S  makes  an  obtuse 

7*" 

angle  with  the  current  /  so  long  as  <p  is  comprised  between  o  and  — 
(the  two  vectors  being  in  opposition  when  </>=o),  and  that  it  is  in  quadra- 

ture when  (b=—. 
r     2 


144  SYNCHRONOUS  MOTORS 

The  expression  for  the  mean  electric  power  can  be  easily  obtained, 
according  to  the  usual  rule,  by  multiplying  the  vector  S  by  the  pro- 
jection of  /  on  S  (I  and  S  being  effective  values).  We  then  have 

P=  SI  cos  A  CB 


=  I2aj  cos  (7r-0-d)/'-fsm2    +    lf~      cos 


on  the  assumption  that 


The  value  of  the  power-factor,  obtained  from  the  preceding,  is 

COS(TT  —  (p  —  d)=  —cos  <[>  cos  d+sin  (p  sin  d 
—  X  cos  (p  sin  (p 


—  X  sin 
or 


4 

'  "2  sin 


whence  P=  — &>/72  sin  6  cos  6=  - 


The  power  is  negative  and  therefore  the  alternator  operates  as  a 
motor  (consuming  electrical  energy  and  producing  mechanical  power) 

when  the  value  of  <p  is  comprised  between  o  and  —  ;  and  it  operates 
as  generator  (producing  electrical  energy  and  consuming  mechanical 
power)  for  values  of  <p  comprised  between  o  and  --  ,  i.e.,  when  the 

current  is  leading. 

The  power  is  a  maximum  for 


and  its  absolute  value  is 


MOTORS  OPERATING  SYNCHRONOUSLY  WITHOUT  D.  C.  145 

This  power  is  necessarily  very  low,  since  ^  is  never  more  than  a 
fraction  of  the  mean  self-induction  /',  and  since  the  E.M.F.  which  is 

utilized, ,  is  itself  only  a  small  fraction  of  the  E.M.F.  of  self-induction 

of  the  alternator,  which  latter  is  always  lower  than  the  E.M.F.  that 
would  be  generated  if  the  alternator  were  excited  by  its  fields.  It 
can  be  easily  seen  that  the  power-factor  always  remains  very  low. 
Moreover,  this  motor  has  the  same  disadvantage  as  polyphase  reaction- 
motors.  For  this  reason,  this  method  of  using  alternators  does  not 
appear  susceptible  of  practical  application. 

Synchronous  Motor  with  Alternating  Fields.  The  invention  and 
the  theory  of  a  synchronous  motor  with  alternating  fields  are  due  to 
Galileo  Ferraris  (see  Memoir  presented  to  the  Academy  of  Sciences  of 
Turin,  Dec.  1893,  and  also  Industrie  Electrique,  10  Juin,  1894).  In 
this  apparatus  the  fields  and  armature  are  excited  in  series  or  in  shunt 
by  alternating  currents  taken  from  the  same  source,  and  the  armature 
is  previously  brought  to  a  speed  double  that  which  would  be  suitable 
for  a  synchronous  motor  with  constant  fields.  The  motor  then  runs 
synchronously,  and  it  can  be  lightly  loaded. 

The  operation  of  such  a  motor  is  easily  explained  by  the  aid 
of  the  theorem  of  Leblanc.  Let  2p  equal  the  number  of  poles  and 
cu  the  speed  of  pulsation  of  the  currents.  The  alternating  field 
(the  field  winding  being  supposed  stationary)  can  be  expressed  ficti- 
tiously by  two  fields,  HI,  H2,  one  rotating  to  the  right,  the  other  to  the 

left,  with  the  contrary  speeds  +—  and .     A  speed  equal  to —  is 

given  to  the  armature  in  one  direction;  the  alternating  field  which  the 
armature  produces  is  itself  also  equivalent  to  two  revolving  fields 

having  the  velocities  H —  and relatively  to  the  armature.     The 

corresponding  absolute  speeds  will  be 


First  field,  Mlt       — +-=^  to  the  right. 
P     P      P 

Second  field,  M2, =—  to  the  right. 

P      P     P 

The  field  MI  will  have  no  action  on  the  fields  HI  and  H2,  whose 
speeds  are  either  different  or  contrary.     Likewise,  M2  will  have  no 


146  SYNCHRONOUS  MOTORS 

action  on  HI.     On  the  contrary,  H2  and  M2,  both  rotating  to  the 
right  with  the  same  speed  — ,  will  produce  a  regular  torque. 

A  motor  of  this  type  can  be  started  more  or  less  well  by  the  methods 
of  starting  already  described  for  ordinary  synchronous  motors.  If 
the  motor  is  started  by  means  of  an  induction-motor  for  two-phase 
currents,  it  is  well  to  group  the  stator-coils  so  as  to  reduce  the  number 
of  poles  by  one-half,  and  thus  obtain  the  double  speed  which  is  necessary. 

A  disadvantage  of  this  system  is  that  it  gives  rise  to  upper  harmonics 
in  the  induced  E.M.F.  of  the  motor.  Moreover,  the  field  MI  of 
the  armature  will  induce  a  harmonic  of  frequency  $ojt  in  the  field- 
circuit,  while  the  fields  will  give,  in  the  armature,  an  E.M.F.  of  the 
form 

A 

A  sin  cot  sin  2ojt=  —  (cos  wt  —  cos  ^cof), 

i-.e.,  also  a  harmonic  of  the  same  frequency,  ^tot. 

Besides,  in  consequence  of  the  high  self-induction,  the  power- 
factor  is  necessarily  very  low. 

;  Another  type  of  single-phase  synchronous  motor  with  alternating 
field  can  also  be  obtained  by  employing,  in  the  armature,  currents 
commutated  by  means  of  a  shell-commutator  analogous  to  that  used 
for  the  Ganz  motors  previously  described.  When  once  synchronism 
has  been  attained,  the  effect  of  this  commutator  is  to  invert  the  current 
in  the  armature  so  that  the  torque  resulting  from  the  action  of  the  field 
on  the  armature  always  retains  the  same  sign.  This  motor  is  about 
equivalent  to  the  motor  with  pulsating  field,  since  the  two  elements 
are  simply  inverted  and  the  armature  can  be  considered  as  a  revolving 
field.  The  only  difference  results  from  the  relative  importance  of  the 
ampere-turns  of  the  armature  in  the  "fields. 


CHAPTER  VII 

BIPOLAR   DIAGRAM  OF  THE  SECOND  KIND   IN 
AMPERE-TURNS 

By  Prof.  C.  A.  ADAMS,  S.B.,  E.E. 

INTRODUCTION 

In  order  to  make  plain  the  transition  from  Fig.  27,  which  is 
fundamentally  an  E.M.F.  diagram,  to  the  corresponding  M.M.F. 
Diagram,  it  will  be  necessary  to  develop  what  may  be  called  the 
General  Vector  Diagram  of  the  Synchronous  Motor. 

Referring  to  Fig.  72,  R  represents  in  magnitude  and  space  phase l 
or  direction,  the  resultant  of  the  field  M.M.F.  F  and  the  armature 
M.M.F.  A.  The  space  phase  of  F  is  its  direction  reduced  to  a  two- 
pole  equivalent;  i.e.,  the  direction  of  the  axis  of  the  field  poles  in 
two-pole  diagram.  The  space  phase  of  A  is,  for  a  single-phase 
machine,  the  average  direction  (with  respect  to  field  structure)  of 
the  armature  M.M.F.,  and  coincides  with  the  direction  of  this 
M.M.F.  when  the  current  is  a  maximum,  right-handed  or  clock- 
wise rotation  of  the  armature  being  assumed.  In  a  polyphase 
machine  the  resultant  of  the  M.M.F.'s  of  the  several  armature 
phases  is  approximately  constant  in  magnitude  and  direction  for  all 
parts  of  a  revolution.  The  time  phase  of  the  armature  current  7 
may  also  be  represented  by  the  direction  of  A',  and  by  choosing  a 
proper  scale  of  amperes,  the  vectors  7  and  A  may  be  made  identical. 

Assuming  that  the  reluctance  of  the  magnetic  circuit  through  the 
armature  and  air-gap  is  the  same  in  all  radial  directions,  there  will 
result  a  flux,  3>,  in  the  same  direction  as  R.  The  rotation  of  the 
armature  through  this  flux  will  cause  to  be  induced  an  E.M.F.  TV  in 
magnitude  proportional  to,  and  in  time  phase  90°  behind,  <£.  The 

1  A  similar  analysis  for  the  alternator  is  given  in  greater  detail  in  the  Harvard 
Engineering  Journal  for  January,  1903.' 

147 


148 


SYNCHRONOUS  MOTORS 


vector  £2'  also  represents  (on  the  space  phase  diagram  with  right- 
handed  l  armature  rotation)  the  direction  of  the  axis  of  an  armature 
coil  when  the  induced  E.M.F.  is  a  maximum,  just  as  the  vector  / 
represents  both  the  time  phase  of  the  armature  current  and  the  direc- 
tion of  the  axis  of  an  armature  coil  when  its  current  is  a  maximum, 
which  is  thus  the  direction  or  space  phase  of  the  armature  M.M.F.  A. 
The  vector  EI  drawn  equal  to  E%  and  in  opposite  phase,  represents 
that  part  of  the  impressed  E.M.F.  E\  necessary  to  balance  the  induced 
E.M.F.  (Ed).  The  impressed  E.M.F.  (Ei)  must  then  be  equal  to 
EI  plus  the  E.M.F.  Ir,  in  phase  with  /,  consumed  by  the  armature 


FIG.  72. 


resistance,  and  the  E.M.F.  Ix,  90°  ahead  of  /,  consumed  by  the 
leakage  reactance  of  the  armature. 

The  diagram  of  Fig.  72  gives  a  fairly  complete  representation  of 
the  relations  involved  in  the  operation  of  the  synchronous  motor. 
Two  assumptions  are  involved,  however,  which  should  be  kept  in 
mind  in  case  of  a  quantitative  analysis :  first,  it  is  assumed  that  the 
flux  <1>  is  in  the  same  direction  as  the  resultant  M.M.F.  R;  but  in  the 

1  It  will  be  observed  that  the  right-handed  or  clockwise  armature  rotation 
here  assumed  for  the  space  phase  interpretation  of  the  diagram  corresponds  to 
the  customary  left-handed  vector  rotation  in  the  time  phase  diagram.  In  the 
case  of  a  revolving  field  type  of  machine,  the  field  must  revolve  left-handedly, 
in  order  to  give  the  same  relative  motion  between  armature  and  field. 


BIPOLAR  DIAGRAM  149 

salient  pole  1  machine  the  reluctance  of  the  magnetic  circuit  through 
the  armature  is  least  in  the  direction  of  F  and  very  much  larger  at 
right  angles  thereto;  the  flux  will  therefore  tend  to  lie  inside  of  R, 
i.e.,  on  the  side  towards  F:  second,  F  is  taken  as  the  total  M.M.F. 
of  the  field  coils  per  complete  magnetic  circuit;  but  a  part  of  F  is  con- 
sumed by  the  reluctance  of  the  field  cores  and  yoke,  and  is  thus  not 
available  to  compound  with  A  in  the  armature  space.  The  errors  due 
to  these  two  assumptions  partly  neutralize  each  other  as  far  as  the 
direction  of  <f>  is  concerned,  since  the  second  throws  it  too  far  out 
and  the  first  tends  to  bring  it  back. 

But  even  this  diagram  with  its  approximations  is  too  complex 
to  be  of  service  in  getting  a  clear  bird's-eye  view  of  synchronous 
motor  operation,  although  it  is  an  excellent  starting  point  for  further 
approximations,  and  will  prove  very  useful  for  reference  purposes. 

The  two  familiar  approximations  are  obtained  as  follows: 

DIAGRAM  TRANSFORMATIONS 

In  any  dynamo-electric  machine  there  are  in  general  two  M.M.F.'s, 
those  of  the  field  and  armature.  In  computing  the  E.M.F.  of  such 
a  machine,  it  is  possible  either  to  consider  the  resultant  of  these  two 
M.M.F.'s  and  the  corresponding  flux,  or  to  consider  the  fluxes  which 
would  be  produced  by  the  two  M.M.F.'s  if  acting  separately,  and  to 
find  the  resultant  of  these  two  fluxes.  The  results  of  the  two  methods 
of  procedure  are  identical  only  when  the  reluctances  of  the  magnetic 
circuits  in  which  the  several  M.M.F.'s  act  are  equal.  This  condition, 
though  never  present  except  in  a  very  roughly  approximate  degree, 
is  nevertheless  a  very  convenient  one  to  assume  for  purposes  of 
approximate  analysis.  With  this  assumption,  the  various  fluxes  in 
any  case  will  be  proportional  to  their  several  M.M.F.'s  and  in  the 
same  direction  or  space  phase;  moreover,  at  constant  speed  of  rota- 
tion the  E.M.F.'s  generated  in  the  armature  by  cutting  through 
these  fluxes  will  be  proportional  thereto  and  in  space  and  time  quadra- 
ture therewith. 

It  will  be  legitimate,  therefore,  under  the  above-mentioned 
assumption,  to  substitute  for  any  M.M.F.  its  corresponding  flux  or 
E.M.F.,  or  to  substitute  for  any  E.M.F.  its  corresponding  flux  or 

1  The  most  accurate  diagram  for  the  salient  pole  machine  is  Professor  Blondel's 
Two-reaction  Diagram  employed  in  Chapter  III;  but  this  does  not  lend  itself 
readily  to  a  simple  visualization  of  the  problem  in  hand,  or  to  simple  calculations. 


150 


SYNCHRONOUS  MOTORS 


M.M.F.,  since  these  are  proportional  to  each  other  in  magnitude  and 
bear  to  each  other  a  fixed  phase  relation. 

E.g.,  in  Fig.  72  there  are  shown  vectors  which  represent  three 
M.M.F.'s,  A,  F,  and  their  resultant  R,  each  one  of  which  would, 
under  the  above  assumption,  produce  a  flux  of  the  same  phase  and 
proportional  in  magnitude.  These  fluxes  &A&F  and  their  resultant 
<£  are  shown  in  Fig.  73,  as  well  as  the  corresponding  E.M.F.'s  EA', 
Ep,  and  £2',  which  would  be  induced  by  the  cutting  of  the  armature 
conductors  through  these  fluxes,  and  which  are  90°  behind  their 
respective  fluxes.  There  are  thus  three  M.M.F.'s,  three  fluxes,  and 
three  E.M.F.'s,  each  set  being  proportional  to  the  other  two,  and 


FIG.  72. 

the  three  triangles  representing  the  relations  between  the  elements 
of  the  three  sets  being  similar. 

E.M.F.  Diagram 

In  Fig.  73  Ep,  EA,  and  E\  are  the  parts  of  the  impressed  E.M.F. 
consumed  by  the  induced  E.M.F.'s  Ep,  EA',  and  £2'  respectively, 
just  as  Ir  and  Ix  are  the  parts  of  the  induced  E.M.F.  consumed  by 
resistance  and  leakage  reactance.  Moreover,  since  EA  is  propor- 
tional to  and  in  quadrature  with  A  (and  therefore  7),  it  is  in  phase 
with  Ix  and  may  be  written, 


=  Ix 


A, 


where  XA  (the  constant  of  proportionality)  is  an  equivalent  reactance 
representing  the  armature  M.M.F.     The  sum  X+XA  =  X  is  called 


BIPOLAR  DIAGRAM 


151 


the  synchronous  reactance,  and  the  corresponding  impedance 
Z=\/r2i-X2  is  called  the  synchronous  impedance.  The  use  of 
these  quantities  reduces  the  synchronous  motor  or  alternator  to  a 
very  simple  basis,  although  at  the  expense  of  accuracy.  .  .- 

It  is  then  possible  to  subdivide  the  impressed  E.M.F.  EI  into 
three  components;  Ep>,  necessary  to  balance  the  counter  E.M.F.  Ep 
which  is  due  to  the  excitation' F  considered  as  acting  alone;  Ir  con- 
sumed by  resistance;  and  IX  consumed  by  the  synchronous  react- 
ance X:  or,  the  synchronous  impedance  E.M.F,  IZ  may  be  looked 


FIG.  73- 

upon  as  the  resultant  of  the  impressed  E.M.F.  EI  and  the  counter 
E.M.F.  EF'.  This  is  shown  more  clearly  in  the  simplified  E.M.F 
diagram  of  Fig.  74,  where  E2  corresponds  to  the  Ep  of  Fig.  73,  and 
E=IZ.  The  current  /will  then  lag  behind  the  resultant  E.M.F.  E 

.    X 
by  an  angle  whose  tangent  is  — . 

The  transformation  from  Fig.  72  to  Fig.  73  consists  in  substituting 
for  the  armature  M.M.F.  A,  an  E.M.F.  EA',  which  would  be  induced 
in  the  armature  by  cutting  the  flux  <E%  which  in  turn  would  be 
produced  by  the  M.M.F.  A  if  acting  in  a  magnetic  circuit  of  the  same 
reluctance  as  that  in  which  R  produces  <£. 


152  SYNCHRONOUS  MOTORS 

The  triangle  E\E<2.E  of  Fig.  74  is  the  E.M.F.  equivalent  of  the 
M.M.F.  triangle  R\FA\  of  Fig.  73;  or,  if  r  and  x  be  neglected,  it 
would  reduce  to  the  E.M.F.  equivalent  of  the  triangle  RFA  of 

Fig.  73- 

This  is  the  basis  of  the  diagrams  of  Chapter  II,  which  are  essen- 
tially E.M.F.  diagrams. 

The  M.M.F.  or  ampere-turn  diagram  is  similarly  obtained  from 
the  general  diagram  by  substituting  for  the  E.M.F.'s  Ir  and  Ix  their 
equivalent  M.M.F.'s  Nir  and  Nix\  i.e.,  Nir  and  Nix  are  the  M.M.F.'s 
which  would  produce  the  fluxes  3>r  and  $x,  the  cutting  of  which  by 
the  armature  conductors  would  induce  E.M.F.'s  equal  and  opposite 
to  Ir  and  Ix  (see  Fig.  73),  where  RI  and  3>i  are  the  M.M.F.  and 
flux  corresponding  to  the  impressed  E.M.F.  EI,  and  A\  the  M.M.F. 


FIG.  74. 

corresponding  to  the  E.M.F.  IZ.  Thus  the  M.M.F.  rectangle  A\R\F 
is  similar  and  equivalent  to  the  E.M.F.  rectangle  IZE\EFj  and  the 
M.M.F.  triangle  A\FR\  of  Fig.  75  is  similar  and  equivalent  to  the 
E.M.F.  triangle  EE2Ei  of  Fig.  74. 

It  is  thus  evident  that  with  constant  impressed  E.M.F.,  constant 
load  and  variable  excitation,  the  point  B,  Figs.  75  and  76,  will  lie 
on  the  arc  of  the  circle  as  does  A 2  in  Fig.  27;  but  it  will  be  more 
interesting  to  go  back  to  fundamental  principles  and  to  make  this 
proof  independent  of  that  for  Fig.  27. 

In  any  direct-current  dynamo-electric  machine  the  electro- 
magnetic torque  is  proportional  to  the  product  of  the  ampere-turns 
on  the  armature  and  the  flux  upon  which  the  armature  current  reacts. 
In  an  alternating  current  machine  the  space  phase  of  the  armature 


BIPOLAR    DIAGRAM 


153 


current  also  enters.  E.g.,  in  Fig.  75  the  armature  current  in  every 
coil  reaches  its  maximum  value  when  the  axis  of  the  coil  has  the 
direction  of  the  vector  A  or  /;  i.e.,  when  the  plane  of  the  coil  is  in 
the  line  Ob,  which  differs  from  the  direction  of  the  flux  $  by  the 
angle  <]/.  Thus  the  torque-producing  effect  of  the  current  is  reduced 
from  its  maximum  possible  value,  in  the  ratio  cos  t/,  and  the  torque  is 


FIG.  75. 
where  &r  is  the  constant  of  proportionality.     But 


10  tf 


where  (R  is  the  reluctance  of  the  magnetic  circuit  and  may  for  the 
present  be  assumed  constant. 

Then  the  output  (which  is  proportional  to  the  torque  at  constant 
speed)  may  be  written 


(i) 


154  SYNCHRONOUS  MOTORS 

where  k%  is  a  constant.     But, 

R  cos  fy'=Ri  cos  <]>—  A\  cos  y, 
where  ty  is  the  phase  difference  between  E\  and  /,  and 

cos  ^  =  NiT-^Ai  =  r-^Z 

also  A  i  is  proportional  to  A  (on  the  assumption  of  constant  reluctance 
in  all  directions  through  the  armature,  see  "  Diagram  Transforma- 
tions," page  149,  etc.,  therefore 

P2  =  K2(AiRicos<]>-Ai2  cos*{),       ....     (2) 

where  K2  is  the  new  constant  of  proportionality. 

Referring  now  to  Fig.  76,  the  isosceles  triangle  OAC  is  constructed 

on  Ri  as  base,  with  y(  =tan-1  —  —  -^=tan-1-  )  as  the  base  angle. 

\  J\  If  T  I 

The  notation  is  the  same  as  in  the  preceding  figures. 

Locate  the  point  B  by  the  rectangular  coordinates  u  and  v  measured 
along  the  axes  OC  and  OD  (perpendicular  to  OC).  Then  u  =  A\  cos  fy, 
Ai2=u2-\-i?  and 


.     .....     (3) 

2  j 

or 

P2  cos 


This  equation  has  exactly  the  same  form  as  equation  (See  Ch.  II, 
near  Eq.  (19))  and  its  interpretation  is  likewise  the  same.  The  radius 
of  the  constant-power  circles  is 

T>   j-  I       // -^A2     ^2  cos  y  ,  , 

Radms= \[ — — - (5) 

cos  Y  \  \  2  /  K2 

The  general  interpretation  of  Fig.  76  is  identical  to  that  of  Fig.  27, 
but  the  method  of  quantitative  application  is  not  quite  so  obvious. 
The  simplest  method  is  as  follows : 

Since  AI,  F,  and  RI  are  measured  in  ampere- turns,  they  may  in 
any  given  case  be  expressed  in  terms  of  the  equivalent  field  amperes, 
RI  being  the  field  current  taken  from  the  saturation  curve  at  the 
point  corresponding  to  the  impressed  E.M.F.  EI,  F  being  the  actual 
operating  field  current,  and  A\  the  field  current  taken  from  the 


BIPOLAR  DIAGRAM  155 

short-circuit  curve  at  the  point  corresponding  to  the  actual  armature 
current  7.  The  reason  for  this  last  step  is  obvious  when  it  is  recalled 
that  when  the  machine  in  question  is  operated  as  generator  on  short 
circuit,  EI  and  J^i  are  zero  and  A\  =  F.  This  is  in  fact  the  definition 
of  AI. 

Equation  (5)  gives  the  radii  of  the  constant  power  circles  in 
ampere- turns;  but  it  is  somewhat  simpler  to  compute  these  radii 
from  the  formula  given  for  the  E.M.F.  diagram,  Eq.  (19),  Chapter  II. 
This  formula  may  be  written 

Radius  (volts)  =  Z+   (  —  }  — -2,    .......     (6) 


or  in  armature  amperes, 


Radius  (armature  amperes)  =  *l  I  —  j  --  -.     .      .     (7) 

Designate  by  kaf  the  ratio  of  short-circuit  armature  amperes  to  the 
corresponding  field  amperes  as  taken  from  the  short-circuit  curve. 
Then  the  radius  in  field  amperes  is 


Radius  (field  amperes)  =  — -  /  (  —  j -.     .     .     . 


(8) 


156  SYNCHRONOUS  MOTORS 

When  P2  =  O  the  circle  passes  through  O  and  A,  and  the  corre- 
sponding radius  is, 

Radius  (field  amperes,  P2  =  0)= -=CO. 


This  together  with  jRi  determines  the  isosceles  triangle  of  Fig.  76, 
although  the  base  angles  0  are  the  same  as  for  the  E.M.F.  diagram 
(Fig.  27)  and  can  be  computed  from  r  and  Z,  the  latter  being  deter- 
mined in  the  usual  manner  from  the  saturation  and  short-circuit 
curves.  The  two  methods  of  constructing  this  diagram  give  the 
same  results  when  the  value  of  Z  employed  is  that  corresponding  to 
the  point  E\  on  the  saturation  curve. 

Thus  the  ampere-turn  diagram  is  completely  and  easily  deter- 
mined by  the  same  information  required  by  the  E.M.F.  diagram; 
namely,  the  armature  resistance,  the  saturation  and  short-circuit 
curves,  and  the  impressed  E.M.F. 

In  the  case  of  the  polyphase  machine,  P%,  EI,  and  r  should  desig- 
nate the  per-phase  values. 

The  ampere-turn  or  M.M.F.  diagram  has  several  advantages  over 
the  E.M.F.  diagram. 

First,  the  substitution  of  an  equivalent  M.M.F.  for  the  leakage 
impedance  E.M.F.,  involves  less  error  than  the  substitution  of  an 
equivalent  E.M.F.  for  the  armature  M.M.F.,  because  in  the  common 
type  of  synchronous  motor  the  armature  M.M.F.  is  large  as  com- 
pared with  the  leakage  impedance  E.M.F. 

Second,  in  determining  the  excitation  or  phase  characteristics 
(Fig.  79)  by  means  of  the  M.M.F.  diagram,  F  is  given  directly 
in  field  amperes,  the  quantity  actually  measured,  and  A\  is 
given  in  equivalent  field  amperes,  for  which  the  corresponding  armature 
current  can  be  taken  directly  from  the  short-circuit  curve,  which  is 
usually  a  straight  line;  whereas  when  taken  from  the  E.M.F.  diagram 
the  results  are  in  terms  of  armature  current  and  of  £2,  the  latter 
being  a  hypothetical  E.M.F.,  which  would  be  induced  if  there  were 
no  armature  reaction,  and  assumed  constant  for  constant  excitation. 

Third,  the  M.M.F.  method  gives  a  clearer  picture  of  the  actual 
phenomena.  E.g.,  if  all  the  constants  refer  to  two  similar  machines 
(generator  and  motor)  together  with  the  connecting  line,  and  if 
these  machines  are  of  the  revolving  field  type,  RI  and  F  are  respect- 
ively the  field  M.M.F.'s  of  generator  and  motor,  and  their  relative 


BIPOLAR  DIAGRAM 


157 


directions  show  clearly  the  relative  angular  positions  of  the  two 
revolving  fields,  6  being  the  coupling  angle  as  before. 

Also  the  torque-producing  mechanism  (the  flux  represented  by 
RI,  the  armature  ampere- turns  represented  by  A\,  and  their  phase 
difference)  is  a  little  more  obvious. 

In  both  E.M.F.  and  M.M.F.  diagrams  the  line  OB  is  in  length 
proportional  to  the  armature  current,  and  the  angle  <J*  designates 
the  phase  difference  between  armature  current  and  impressed  E.M.F. 


4eo 


440 


400 


SCO 


320 


tf 


8         10        12        14 
Field  Amperes 


16        18        20        22 


FIG.  77. 

In  the  case  of  a  normal  machine,  the  armature  resistance  is  small 
as  compared  with  the  synchronous  reactance  X,  and  the  constant- 
power  circles  flatten  out  almost  to  straight  lines  (see  Fig.  78). 

Example.  In  Fig.  77  are  shown  the  saturation  and  short-circuit 
curves  of  a  three-phase  60  H.P.  44o-volt  synchronous  motor. 

The  M.M.F.  diagram  is  shown  in  Fig.  78  and  the  phase  charac- 
teristics in  Fig.  79.  If  the  abscissae  of  the  phase  characteristics, 
determined  from  the  E.M.F.  diagram,  were  to  be  transformed  from 
E.M.F.  to  field  current  through  the  medium  of  the  saturation  curve, 


158 


SYNCHRONOUS  MOTORS 


the  results  would  be  found  to  differ  widely  from  the  curves  of  Fig.  79, 
the  latter  being  much  more  accurate. 


r=0.2      X  =  4.0 
(  per  delta  phase) 

,100jLPi___-  - 

RO  H.P. 

Ai/'\vp 

«n  H.P.          — 

(~1^/                    ^^ 

An  H.P.         . 

_/                                       ^\ 

on  W.P.           _ 

Rl  =  14.0  Amps.           """^^k 

FIG.  78. 


Approximate  Diagram 

If  the  resistance  r  be  neglected  in  Fig.  76,  OC  and  AC  will  become 
vertical  and  parallel,  and  the  constant-power  circles  will  degenerate 
into  straight  lines  (see  Fig.  81). 

Following  the  interpretation  of  Figs.  27  and  (76)  we  have,  as  the 
general  interpretation  of  Fig.  81 : — 


wioo 
I  ^ 

<5 
|   60 


\\ 


\ 


Lagging 


Lea 


ing 


10  15 

Field  Amperes 

FlG.  79. 


20 


25 


The  region  to  the  right  of  the  vertical  line  55  corresponds  to 
unstable  operation,  while  the  region  to  the  left  corresponds  to  stable 
operation. 

The  region  above  OD  and  between  LL  and  55  corresponds  to 


BIPOLAR   DIAGRAM  159 

lagging  motor  current,  while  that  above  the  line  OD  and  to  the  left 
of  LL  corresponds  to  leading  motor  current. 

The  region  below  OD  and  between  LL  and  SS  corresponds  to 
leading  generator  current,  while  that  below  OD  and  to  the  left  of  ZZ 
corresponds  to  lagging  generator  current. 

Extreme  Cases 

The  assumptions  (involved  in  the  above-described  diagrams)  that 
the  flux  <£  has  the  same  direction  as  R,  and  that  constant  <£  means 
constant  7?,  irrespective  of  the  direction  or  space  phase  of  <£,  become 


fjfc 


FIG.  80. 

less  and  less  warrantable  as  the  conditions  of  operation  depart  farther 
and  farther  from  the  normal. 

E.g.,  consider  the  case  of  light  load  and  very  low  excitation 
(Fig.  80).  R  makes  such  a  large  angle  with  F  (the  axis  of  the  field 
magnets)  that  $  will  not  only  be  less  in  proportion  to  R,  owing  to 
the  greater  reluctance  of  the  magnetic  circuit  in  this  direction,  but 
it  will  also  be  pulled  down  more  nearly  into  the  direction  of  F,  because 
that  is  the  direction  of  minimum  reluctance.  Then  E\  (=Ei)  will 
also  be  shifted  by  an  equal  angle,  since  it  must  be  in  quadrature 
with  $.  This  is  shown  by  the  light  lines  of  Fig.  80.  Thus  R  will 
lag  behind  EI  by  less  than  90°,  and  if  F  be  reduced  to  zero,  A  (=R) 
will  still  have  a  torque  component  and  the  motor  will  operate  under 
very  light  load  without  any  excitation. 


160 


SYNCHRONOUS  MOTORS 


The  physical  explanation  of  this  is  that  the  armature  current 
reacts  upon  a  flux  produced  wholly  by  said  armature  current.  Ordi- 
narily this  self-produced  flux  would  be  perpendicular  to  the  plane  of 
the  coil,  and  would  therefore  have  no  component  in  the  plane  of  the 
coil  upon  which  the  current  could  react  to  produce  a  torque;  but  in 
this  case  the  unsymmetrical  reluctance  of  the  magnetic  circuit  results 
in  a  flux  not  in  the  same  direction  as  the  M.M.F.,  and  the  motor 
will  actually  run  without  excitation  at  very  light  load.  The  arma- 
ture current  is,  of  course,  excessive. 


FIG.  81. 

Mechanical  Analogue 

The  approximate  diagram  of  Fig.  81  lends  itself  readily  to  the 
explanation  of  a  very  interesting  mechanical  analogue  to  the  synchro- 
nous motor. 

Consider  two*shafts,  a  driver  and  a  driven,  placed  end  to  end  in 
the  same  line  and  connected  together  by  an  elastic  coupling  consisting 
of  an  elastic  string  or  band,  the  two  ends  of  which  are  connected  to 
two  crank  pins  on  the  two  adjacent  ends  of  the  shafts  in  question. 
Assume  further  that  the  tension  of  the  elastic  band  is  directly  pro- 
portional to  the  distance  between  the  two  crank  pins  which  lie  in 
the  same  plane  perpendicular  to  the  shafts.  In  Fig.  82  is  shown  a 


BIPOLAR  DIAGRAM 


161 


diagrammatic  end  view  of  this  coupling:  the  point  D  is  the  center 
of  the  shaft;  the  line  DO  is  the  radius  of  the  driving  crank  pin  and 
DB  that  of  the  driven  crank  pin;  OB  is  thus  the  distance  between 
the  two  crank  pins,  to  which  the  tension  of  the  elastic  band  is  pro- 
portional; the  lever  arm  of  the  tension  about  the  center  of  the  shafts 
is  DC  =  DB  cos  cj>,  and  the  corresponding  turning  moment  trans- 
mitted from  driver  to  driven  is  thus  proportional  to  DCXOB.  But 
T>C = DO  cos  <|>,  and  the  torque  or  turning  moment  is  T=OBXlX) 
cos  <J;;  or  since  OB  cos  fy  =  OBo,  T=OBoXOD',  i.e.,  the  torque  is  pro- 
portional to  the  area  of  the  rectangle  DOBo.  OBo  is  the  tangential 


FIG.  82. 

component  of   the   tension  OB,  and  OD  is  the  corresponding  lever 
arm. 

Consider  now  the  operation  of  this  coupling  for  given  crank  pin 
radii  and  a  given  stiffness  of  the  elastic  band  OB.  At  no  load  or 
zero  torque  the  crank  DB  will  be  pulled  ahead  into  line  with  DO, 
the  tangential  component  of  the  tension  OB  will  be  zero,  the  angle  0 
will  be  zero,  BQ  will  coincide  with  O,  and  the  rectangle  DOBo  will 
collapse  into  a  line  according  to  the  above-indicated  relation  between 
the  torque  and  the  area  of  this  rectangle.  If  now  the  load  torque 
be  increased,  the  point  B  will  fall  back  to  the  right,  and  the  angle  0 
will  increase  until  OBo,  the  tangential  crank  effort,  has  attained  the 
the  value  imposed  by  the  load.  A  further  increase  in  the  load  torque 
will  be  accompanied  by  a  further  increase  in  0  and  in  OBo;  but  it 


162  SYNCHRONOUS  MOTORS 

will  be  observed  that  there  is  a  limit  to  the  possible  increase  of  OBo, 
namely,  when  6  is  90°  and  OBo^DB,  beyond  which  the  tangential 
crank  effort  OBo  decreases,  to  zero  when  6=180°,  to  a  negative 
maximum  when  6=270°,  and  back  to  zero  again  when  6  =  360°  or  o°. 
Thus  although  the  tension  OB  goes  on  increasing  after  6  has  passed 
the  90°  point,  the  lever  arm  decreases  more  rapidly  than  the  tension 
increases,  i.e.,  the  tangential  component  (OBo)  of  the  tension  decreases. 

Assume  now  that  with  constant  load-torque  the  driven  crank  pin 
radius  is  altered.  The  tangential  crank  effort,  OBo,  must  remain  the 
same,  and  the  driven  crank  pin  B  must  lie  somewhere  on  the  line 
BoBoo-  Thus  for  a  given  driver  crank  radius,  the  angle  6  and  the 
tension  OB,  depend  upon  the  torque  demanded  by  the  load  (which 
is  proportional  to  OBo),  and  upon  the  driven  crank  pin  radius  DB. 
For  each  value  of  the  load  torque  and  of  OBo,  the  tension  will  be  a 
minimum  for  that  value  of  the  driven  crank  pin  radius  DB,  which 
causes  B  to  fall  at  BQ.  It  is  obvious  that  this  particular  value  of 
DB  varies  with  the  load. 

If  now  the  driver  crank  pin  radius  DO  be  increased,  the  value  of 
OBo  for  a  given  load  will  be  decreased  and  vice  versa,  since  their 
product  must  be  constant. 

It  has  been  assumed  thus  far  that  a  fixed  relation  exists  between 
the  crank  pin  displacement  OB  and  the  coupling  tension;  but  while 
this  is  true  for  a  given  case,  it  is  evident  that  a  different  elastic  band 
or  spring  could  be  substituted  which  would  have  a  different  elastic 
coefficient,  and  with  which  the  angle  6  would  be  quite  different  for 
the  same  load. 

The  angle  6  will  be  hereafter  referred  to  as  the  coupling  angle. 
It  obviously  depends  upon  the  load  torque,  the  two  crank  pin  radii, 
and  the  elastic  coefficient  of  the  coupling. 

Let  p  =  tension  in  pounds  =  kXOB,  where  k  is  the  coefficient  of  the 
elastic  band.  Then  if  DO  and  DB  are  measured  in  feet,  the  torque 
in  pound-feet  will  be 

T=kXOBoXDO=kXOB  cos  tyXDO=kXDB  sin  6XZX). 

Thus  for  any  given  value  of  the  crank  pin  radii,  the  torque  varies 
as  the  sin  6. 

A  comparison  of  the  diagrams  of  Figs.  81  and  82  will  show  their 
exact  mechanical  equivalence.  In  both  cases  the  torque  is  propor- 
tional to  OBo  for  a  given  value  of  OD',  therefore  the  relation  of  the 
torque  to  6  and  to  DB  is  exactly  the  same  in  the  two  cases.  In 


BIPOLAR  DIAGRAM  163 

Fig.  81  OB  is  a  measure  of  the  armature  M.M.F.,  i.e.,  the  strength 
of  the  armature  considered  as  an  electromagnet  lying  across  the 
magnetic  field  whose  flux  is  <l>  and  whose  M.M.F.  is  R  (see  Fig.  83). 
The  resulting  torque  is  then  proportional  to  A 1$  cos  fy  or  approxi- 
mately proportional  to  A  iR  cos  fy ;  but  in  this  approximate  analysis, 
$  and  therefore  R  is  determined  by  the  impressed  E.M.F.,  and  is 
in  any  given  case  a  rough  measure  of  the  exciting  M.M.F  of  the 
generator  which  supplies  the  power,  while  F  is  the  exciting  M.M.F. 
of  the  synchronous  motor. 

Thus  the  two  crank  pin  radii  correspond  roughly  to  the  exciting 
M.M.F.'s  of  the  generator  (driver)  and  the  motor  (driven),  the 
tension  OB  corresponds  to  the  armature  M.M.F.,  the  angle  fy  to  the 
phase  difference  between  current  and  E.M.F.  or  to  the  angle  by 
which  the  armature  M.M.F.  differs  from  its  position  of  maximum 


FIG.  83. 

torque-producing  effect,  and  the  coupling  angle  0  roughly  to  the 
mechanical  phase  difference  (measured  in  electrical  degrees)  between 
the  revolving  parts  of  the  generator  and  motor. 

The  electrical  angular  velocity  of  the  motor  is  the  same  as  that 
of  the  generator,  but  their  mechanical  angular  velocities  are  inversely 
as  their  numbers  of  poles;  while  in  the  mechanical  analogue  the 
angular  velocities  of  the  two  parts  of  the  coupling  must  be  the  same. 

In  order  to  see  what  determines  the  stiffness  of  the  electro- 
magnetic coupling  between  alternator  and  synchronous  motor,  it  is 
necessary  to  remember  that  the  torque  is  strictly  proportional  to  the 
product  of  the  flux  $  and  that  component  of  the  armature  M.M.F. 
in  quadrature  with  <£,  and  that  for  a  given  value  of  3>  the  necessary 
M.M.F.  R  will  depend  upon  the  reluctance  of  the  magnetic  circuit; 
e.g.,  if  a  synchronous  motor  has  its  pole  faces  bored  back  so  as  to 
increase  the  reluctance  of  the  magnetic  circuit,  a  larger  resultant 


164  SYNCHRONOUS  MOTORS 

excitation  R  must  be  supplied  in  order  to  produce  the  same  flux  <i> 
made  necessary  by  the  same  impressed  E.M.F.;  but  with  this  same 
flux,  the  same  load  will  require  the  same  torque  component  OBo  of 
the  armature  M.M.F.  A.  Thus  although  R  and  F  have  increased, 
OBo  is  unchanged,  and  the  angle  6  reduced,  for  the  same  ratio  of  F 
to  R.  This  decrease  of  the  angle  6  for  a  given  torque  corresponds 
to  a  stiffening  of  the  elastic  band  in  the  mechanical  coupling,  and 
increases  the  break-down  torque  (when  6  =  90°).  In  fact,  the  above- 
described  change  in  the  synchronous  motor  is  commonly  called  a 
stiffening  of  the  field,  in  that  the  latter  is  then  less  distorted  by  a 
given  armature  M.M.F. 

It  will  be  observed  in  this  connection  that  every  synchronous 
motor  has  a  natural  period  of  oscillation  about  its  mean  running 
position,  which  depends  upon  the  moment  of  inertia  of  the  revolving 
part  and  the  stiffness  of  the  electromagnetic  coupling.  These  oscil- 
lations correspond  to  a  variation  of  the  coupling  angle  0  about  its 
mean  value,  and  a  pulsation  of  power-flow.  If  the  angular  velocity 
of  the  supply  alternator  should  oscillate  about  a  mean  value,  the 
resulting  oscillation  transmitted  to  the  synchronous  motor  through 
the  electromagnetic  coupling  would  depend  upon  the  amplitude  and 
frequency  of  the  impressed  oscillation,  upon  the  moment  of  inertia  of 
the  revolving  part  of  the  motor,  and  upon  the  stiffness  of  the  coupling. 
If  the  impressed  oscillation  should  have  the  same  frequency  as  that 
of  the  freely  oscillating  motor,  the  latter  would  tend  to  increase  its 
amplitude  of  oscillation  until  it  would  swing  past  the  point  of  maxi- 
mum torque  (6  =  90°)  and  break  down.  If,  on  the  other  hand,  the 
frequencies  were  widely  different,  the  amplitude  of  the  transmitted 
oscillation  would  be  greater,  the  stiffer  the  coupling  and  the  less  the 
moment  of  inertia  of  the  revolving  part. 

These  phenomena  are  usually  covered  by  the  term  "  hunting  " 
or  "  phase  swinging,"  a  quantitative  analysis  of  which  is  given  in 
Chapter  IV. 

Length  of  Air -Gap 

It  will  be  interesting  here  to  review  the  relative  merits  of  high 
and  low  synchronous  impedance  in  a  synchronous  motor;  i.e.,  of  a 
soft  and  a  stiff  electromagnetic  coupling,  or  what  is  equivalent,  the 
relative  merits  of  a  short  and  a  long  air-gap. 

Take  first  a  system  in  which  there  is  a  considerable  pulsation  of 


BIPOLAR  DIAGRAM  165 

frequency  due  to  the  non-uniform  crank  effort  of  a  single  cylinder 
steam-engine  which  drives  the  supply  alternator. 

If  the  impedance  (including  generator,  motor  and  line)  be  very 
low,  i.e.,  if  the  air-gap  be  long  and  the  coupling  stiff,  the  motor 
speed  will  follow  closely  the  frequency  pulsation.  There  is  thus  a 
pulsation  strain  on  the  coupling  accompanied  by  a  large  pulsation  of 
power  and  current.  If  the  motor  is  large  enough  as  compared  with 
the  alternator,  the  stiff  coupling  will  have  the  effect  of  adding  fly- 
wheel capacity  to  the  alternator  and  will  tend  to  reduce  the  frequency 
pulsation,  but  at  the  expense  of  heavy  current  and  power  pulsations. 
If,  on  the  other  hand,  the  impedance  be  large,  the  frequency 
pulsations  will  be  partly  absorbed  in  the  soft  coupling,  the  motor 
will  not  follow  closely  the  frequency  pulsations,  and  there  will 
be  much  less  pulsation  of  power  and  of  current.  There  is,  how- 
ever, an  obvious  limit  to  the  desirable  softness  of  coupling,  namely, 
that  beyond  which  there  is  danger  of  a  breakdown  of  the  motor 
because  of  the  soft  coupling. 

There  is  also  another  objection  to  the  soft  coupling,  which  will 
appear  from  an  inspection  of  the  E.M.F.  diagram  (Fig.  27).  If  the 
reactance  be  high,  the  coupling  angle  will  be  relatively  large  for  a 
given  power,  and  there  will  be  a  large  variation  in  the  power  factor 
of  the  motor  from  no  load  to  full  load  under  constant  excitation,  or  a 
frequent  adjustment  of  excitation  will  be  demanded  in  order  to 
maintain  a  constant  power  factor. 

When  polyphase  synchronous  motors  or  synchronous  converters 
are  started  by  the  induction  motor  action  of  the  damping  coils,  or 
by  hysteresis  and  eddy  current  torque,  it  is  desirable  to  have  as  short 
an  air-gap  as  possible  in  order  to  keep  the  necessary  starting  current 
within  limits. 

The  choice  of  air-gap  is  thus  a  compromise  and  depends  some- 
what upon  the  particular  service  demanded  of  the  motor  in  question, 
and  upon  the  nature  of  the  system  on  which  it  operates.  The  gap 
should  preferably  be  not  as  long  as  demanded  by  good  regulation  in 
the  synchronous  alternator,  nor  as  short  as  demanded  by  high-power 
factor  in  the  induction  motor  or  induction  generator.  On  this  point 
there  is  considerable  difference  in  practice,  particularly  in  the  case 
of  synchronous  converters. 


CHAPTER  VIII 

GENERALIZATION  OF  DIAGRAM  FOR  COUPLED 
SYNCHRONOUS  MACHINES 

THE  diagram  represented  in  Fig.  27  (page  44),  which  is  based 
upon  Joubert's  theory  that  the  armature-reaction  of  the  alternator 
employed  can  be  adequately  explained  by  supposing  the  presence 
in  the  armature  of  a  simple  mean  synchronous  reactance,  can  be 
retained  to  advantage  for  certain  purposes  even  in  the  case  of  the 
theory  of  "  two  reactions,"  developed  by  the  author  in  Part  III, 
Chapter  I,  for  alternators  with  saturated  magnetic  fields.  It 
may,  indeed,  be  noted  that  the  impedance  Z,  which  enters  into  the 
expression  for  the  vector,  A \A^  is  still  a  constant  of  the  same  char- 
acter as  before,  though  it  now  represents  the  impedance  of  the  trans- 
verse reaction,  which  may  be  symbolized  by  Z*,  when  we  take 


We  will  first  suppose  the  case  of  two  alternators  which  comply 
with  Joubert's  law  (i.e.,  which  have  a  constant  synchronous  react- 
ance). The  clearest  form  of  electrical  transmission  diagram  will  be 
obtained  by  first  combining  the  diagrams  shown  in  Fig.  27  (page  44) 
and  in  Fig.  43  (page  86).  All  that  is  necessary  is  to  point  off  along  OA 
(Fig.  84)  a  distance  proportional  to  the  voltage  (U)  at  the  terminals 
of  the  two  alternators,  these  being  assumed  to  be  near  each  other. 
(If  they  are  far  apart  the  impedance  of  each  one  is  to  be  increased 
fictitiously  by  an  amount  equal  to  half  the  impedance  of  the  line 
by  which  they  are  coupled  together).  The  vectors  of  the  e.m.f.'s 
of  these  alternators  are  then  drawn  in  proper  magnitude  and  phase. 
(The  alternators  may  be  unlike  provided  they  both  have  the  same 

characteristic  ratio  — '  =  tan  y).     We  then  draw  the  line  of  zero 

166 


DIAGRAM  FOR  COUPLED  SYNCHRONOUS  MOTORS     167 

phase-angle,  AN,  lagging  by  the  angle,  y,  with  respect  to  U,  and 
point  off,  at  N,  the  center  of  the  circles  of  constant  internal 
power,  which  will  be  the  same  for  the  generator  as  for  the  motor. 
The  three  points  A\AA^  are  then  joined  by  a  straight  line,  A iAz, 
which  represents  twice  the  product  Z<7,  of  the  impedance  Zt  by 
the  current,  /,  that  is  to  say  twice  the  impedance  e.m.f.  of  the 
circuit.  In  reality  the  e.m.f.  U  may  be  considered  as  being  that  of 
a  "  line  "  absorbing  the  energy  of  the  generator  and  supplying  the 


FIG.  84. 


same  energy  to  the  motor.  We  thus  introduce  a  "  fictitious  "  line 
which  does  not  change  the  phenomena  in  any  way,  but  which  has 
the  advantage  of  enabling  the  diagram  to  be  generalized  completely. 

A  A  - 

The  current  7  passing  between  the  two  alternators  is  equal  to  —^- 

and  it  lags  behind  A  A  \  by  the  angle  y. 

In   the   "  two-reactions  "   theory,   this   diagram  is  modified  in 
the  manner  now  to  be  explained.     The  phases  of  the  currents  are 


168 


SYNCHRONOUS  MOTORS 


still  deduced  from  those  of  the  e.m.f.'s  by  means  of  the  internal 
resistance  r  and  an  inductance;  but  that  inductance  is  then  the  total 
transverse  inductance  Lt  =  lt-\-s.  The  sides,  A2A  and  AA\,  of  the 
triangles  will  therefore  represent  ZJ,  where  Z*  symbolizes  the  impe- 
dance which  corresponds  to  the  transverse  inductance,  where  / 

idl~/t 
represents  the   current,  and   where    tan  y  = .       But   OA\    and 

OA2  no  longer  represent  the  internal  e.m.f.'s  except  in  regard  to 
their  phases.  To  represent  them  in  magnitude  as  well  as  in  phase, 
the  diagrams  shown  in  Fig.  85  must  be  constructed  by  reference  to 
the  characteristic  curves  for  these  two  alternators,  which  curves, 
for  the  sake  of  simplicity,  are  here  supposed  to  be  the  same  for  both 
machines. 


Let  OMiNi  be  the  total  excitation  characteristic  or  saturation 
curve  obtained  by  plotting  the  excitation  ampere-turns  as  abscissas 

and  the  induced  e.m.f.'s   at  no  load  as  ordinates.     Let   -^-=-  =  the 

A/2 

armature  ampere-turns  produced  by  an  effective  current  /  in  the  arma- 
ture (with  N  peripheral  conductors  per  pair  of  poles;  and  let  us  select, 
for  Fig.  84,  a  scale  such  that  the  current- vector  A I  will  also  represent 
these  armature  ampere-turns  according  to  the  scale  of  abscissas  in 
Fig.  85.  We  then  draw  AB\  and  AB2,  perpendicular  respectively 
to  OAi  and  OA2  (in  Fig.  84);  and  we  point  off  AFi  and  AF2  both 
equal  to  cos/,  and  also  draw  FiD\  and  F2D2,  perpendicular  respect- 
ively to  OA i  and  OA2.  The  vectors  OD\  and  OD2  represent  the 


DIAGRAM  FOR  COUPLED   SYNCHRONOUS  MOTORS     169 

internal  e.m.f.'s  generated  in  the  two  alternators  by  the  direct  flux, 
i.e.,  the  flux  along  the  field  axis.  Let  Si  and  82  represent  these 
two  e.m.f.'s,  and  let  them  be  drawn  as  ordinates  on  the  total  charac- 
teristic curve  (Fig.  85).  The  corresponding  abscissas  omi,  and  om%, 
represent  the  excitation  ampere-turns  necessary  to  generate  these 
e.m.f.'s  at  no  load.  We  add  to  them,  respectively,  the  counter 
ampere-  turns  of  the  armature,  m\n\  and  m^n^  obtained,  respectively, 
oy  projecting  (Fig.  84)  the  vector  AI  on  the  perpendiculars  to  OA  \ 
and  OA2,  since  these  projections  represent  the  reactive  components 
of  the  armature  ampere-turns.  They  are 

KN2J2 

and 


V2  V2 

in  which  N\  and  A72  represent  the  numbers  of  peripheral  armature- 
conductors  per  pair  of  poles  in  the  two  machines,  and  K  is  the  utili- 
zation-coefficient for  the  particular  winding.  We  thus  obtain  the 
e.m.f.'s  EI  and  £2;  but  these  are  not  yet  altogether  exact,  because  we 
have  neglected  the  small  increases  of  excitation  which  are  necessary  to 
compensate  for  the  increase  in  saturation  of  the  field  magnet  cores 
resulting  from  the  increase  in  magnetic  leakage  between  the  pole- 
pieces.  The  correction  necessary  is  very  easily  made,  if  the  permeance 
Bj,  of  the  leakage  path  is  known  and  if  the  excitation-characteristic  of 
the  magnetic  field  cores  alone  (OQ)  has  been  drawn,  as  shown  in  Fig. 
85,  by  plotting  it  reversed,  to  the  left  of  the  axis  of  ordinates,  taking 
the  excitation  ampere-turns  as  abscissas  and  the  magnetic  flux 
through  the  magnets  and  yoke  as  ordinates.  This  characteristic  curve 
should  be  drawn  according  to  a  scale  of  ordinates  such  that  the 
magnetic  fluxes  may  be  represented  by  the  electromotive  forces 
which  they  would  generate  in  the  armature  if  they  were  to  thread 
through  it.  Let  (}  be  the  angle  of  the  tangent  to  the  curve  OQ 
at  the  point  c.  Let  us  now  draw  at  M\  an  angle  bMia  =  <x,  whose 
tangent  represents  the  permeance  Bj.  of  the  magnetic  leakage  path. 
Then  the  Vertical  segment  ab,  intercepted  on  the  line  N\n\,  repre- 
sents the  additional  magnetic  leakage  flux.  Let  us  .draw  two  hori- 
zontal lines  ac  and  bd,  and  take  their  intersection  with  the  field- 
characteristic  (at  the  left  in  Fig.  85),  and  let  us  then  find  the  cor- 

tan  a 

responding  abscissas,  p\q\.     The  segment  piqi  =  .  ---  ^m\n\  represents 

tan  p 

and  measures  the  supplementary  ampere-turns  required.     Let  us 


170  SYNCHRONOUS  MOTORS 

take  n\ni=p\qi'y  and  m\n\  represents  and  measures  the  total 
ampere-turns  made  necessary  by  the  armature-current.  In  like 
manner,  m^n^  represents  the  corresponding  ampere-turns  required 
for  the  motor  OAi  at  the  output  indicated. 

To  the  two  values  on'i,  and  on' 2,  of  the  total  ampere- turns,  cor- 
respond electromotive  forces  E',  and  equal,  respectively,  to  Ni'n\ 
and  N2fn2f,  on  the  total  characteristic.  These  e.m.f.'s  (which  are 
those  that  would  appear  if  the  current  /  ceased  suddenly  to  pass 
through  the  alternators),  correspond,  in  the  present  case,  to  what  we 
term  internal  electromotive  forces  in  the  case  of  Joubert's  theory. 
By  measuring  off  (Fig.  84)  from  O,  along  OAz  and  OAi,  distances 
OKi  and  OK2,  respectively  equal  to  these  e.m.f.'s,  we  obtain  two 
vectors  which  replace,  in  the  "  two-reactions  "  theory,  the  vectors 
OA i  and  OA%  in  the  diagram  obtained  with  Joubert's  theory. 

It  is  seen  that,  in  consequence  of  saturation,  these  vectors  are, 
in  general,  shorter  than  OAi  and  OA^  but  this  is  not  a  necessary 
result,  because  it  is  possible  to  design  an  alternator  in  which  the 
transverse  reaction  would  be  very  small,  by  reducing  the  width  of 
the  pole-pieces,  and  which  could,  consequently,  lead  to  a  length  for 
OA  i  less  than  OK\,  at  least  so  long  as  the  saturation  of  the  field- 
magnets  is  low. 

For  the  sake  of  simplicity  Lt  has  been  assumed  constant  at  all 
loads,  because  the  influence  of  the  air-path  is  generally  prepondera- 
ting in  the  transverse  magnetic  circuit,  contrary  to  what  is  true  for 
the  principal  circuit ;  but  those  who  may  wish  greater  refinement  can 
take  into  account  the  slight  variations  of  Lt  with  the  load-conditions 
of  the  alternator,  and  they  can  also  replace  the  perpendicular  straight 
lines  ABi  and  AJ52  by  curves  comprised  between  these  straight  lines 
and  segments  of  circles.  In  this  way  allowance  can  be  made  for 
the  fact  that,  in  certain  cases,  the  transverse  reaction  may  not  raise 
the  voltage  at  the  terminals  as  much  as  when  the  pole-pieces  are 
wide.  This  complication  of  the  diagram  does  not  materially  increase 
its  precision  and  it  is  therefore  preferable  to  use  the  simpler  theoret- 
ical diagram. 

The  circles  of  constant  power  of  the  old  diagram  are  to  be  replaced 
here  by  lines,  which  are  drawn  by  points.  To  obtain  these  lines, 
the  values  of  £'2  are  drawn  or  determined  for  a  large  number  of 
positions  of  A%  and  for  the  corresponding  power-outputs,  and  each 
power- value  is  indicated  near  the  point  A^  corresponding  to  it. 
When  the  entire  diagram  has  been  in  this  way  covered  by  a  sufficiently 


DIAGRAM  FOR  COUPLED  SYNCHRONOUS  MOTORS     171 

numerous  series  of  points,  these  may  be  considered  as  being  "  levels  " 
on  a  topographical  plan,  and  lines  connecting  together  the  points 
at  the  same  "  levels  "  can  be  drawn  like  the  "  contour  "  lines  on 
topographical  plans.  Between  any  two  such  lines  of  constant 
power,  other  lines  of  constant  power  of  intermediate  value  can  be 
drawn  by  interpolation  in  the  well-known  way. 


PART  II 

GENERAL  DIAGRAMS  DEDUCED  FROM   THE    DIAGRAM 
FOR  SYNCHRONOUS  MOTORS 


CHAPTER  I 

GENERAL  DIAGRAMS   DEDUCED   FROM   THE   DIAGRAM  FOR 
SYNCHRONOUS  MOTORS 

Introduction.  The  author  presented,  at  the  Electrical  Congress  in 
Paris,  in  1900,  a  paper  on  The  Graphical  Theory  of  Rotary  Converter 
Regulation,  the  object  of  which  was  to  set  forth  a  purely  graphical, 
but  nevertheless  complete,  method  of  treating  this  complicated  though 
interesting  question,  and  to  show,  at  the  same  time,  that  the  solution 
of  the  problems  involved  could  be  reduced  to  the  solution  previously 
worked  out  for  synchronous  motors,  by  the  author,  and  could  be 
reconciled  with  his  "  theory  of  two  armature-reactions  in  alternators." 

In  accordance  with  that  theory,  the  rotary  converter  can,  in  effect, 
be  treated  as  a  simple  particular  case  of  the  synchronous  motor, 
namely,  as  a  synchronous  motor  having  no  transverse  reactions;  and 
the  vectors  obtained  for  synchronous  motors  are  rendered  applicable 
to  rotary  converters  by  merely  suppressing  the  transverse  reaction 
vector. 

It  is  thus  seen  that  this  theory  contains  nothing  artificial,  but  that, 
on  the  contrary,  it  is  both  general  and  homogeneous  to  a  high  degree. 

Notation.  The  notation  used  may  be  substantially  the  same  as 
for  synchronous  motors;  but  the  meaning  of  some  of  the  subscripts 
must  be  changed  somewhat  to  take  into  account  the  fact  that  the 
armature  of  the  rotary  converter  receives  currents  of  two  kinds,  primary 

172 


GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS    173 

and    secondary.     (The    primary    and    secondary    windings    can    be 
different.)     The  notation  used  will  be  the  following: 

T=  periodic  time  (duration  of  one  period),  in  seconds; 
o>=—=  frequency,  or  speed  of  pulsation  of  the  current; 

R=  total  resistance  of  circuit  (generator,  line,  and  primary  winding  of 

converter) ; l 

1=  cumulative  self-induction  of  generator  and  line; 
s=  self-induction  of  converter  due  to  magnetic  leakage  of  armature; 
L=  self  -induction  added  in  the  form  of  reactance-coils; 
X=a>(L+l+s)  =  total  reactance  of  alternating  current  circuit; 
Z=*v  R2+X2= total  impedance  corresponding; 

E=  effective  E.M.F.  (being  the  E.M.F.  supplied  by  the  line  or  that 
induced  in  the  generator,  in  the  case  of  a  simple  transmission 
circuit).     This   E.M.F.    is   to  be   considered   always   constant 
unless  otherwise  stated; 
q=  number  of  phases  of  the  primary  current; 
e=  effective  primary  (alternating)  E.M.F.  of  the  converter; 

^=E2=  secondary  or  induced  (B.C.)  E.M.F.; 

•» 

6=  ratio  of  transformation  of  the  converter; 

/=  effective  value  of  one  of  the  primary  (alternating)  currents; 
Iw,  or  /,= active  component  of  that  current; 
Id= reactive  component  of  that  current; 

0= phase-angle  of  lag  corresponding  to  ( tan<£= — ) ; 

•*  w 

6= phase-angle  between  the  E.M.F.'s  £  and  E; 
72=  secondary  (direct)  current,  proportional  to  /  less  the  losses; 
N=  number  of  peripheral  primary  conductors  per  magnetic  field; 
K=  reduction-coefficient  for  primary  winding; 

A^2=  number  of  peripheral  secondary  conductors  per  magnetic  field; 
A  =  ampere-turns  of  (shunt)  field-coils; 
n=  number  of  exciting  turns  of  series- winding; 
r=  resistance  per  turn  of  shunt- winding; 
/o  =  active  current  when  the  load  is  zero; 
«o= reactive  current  when  the  load  is  zero. 

1  The  resistance  of  the  line  and  generator  are  negligible  in  the  case  of  a  rotary 
converter  supplied  with  current  from  a  large  distribution-system. 


174    GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

Generalities.  Reduction  of  all  Armature-Reactions  to  the  Single 
Direct  Reaction.  The  inducing  field  is  the  resultant  of  three  magneto- 
motive forces  (Fig.  i): 

1.  The  inducing  ampere-turns  of  the  exciting  current; 

2.  The  ampere-turns  which  are  reactive  with  respect  to  the  E.M.F. 

and  which  act  in  the  same  way  as  the  preceding; 

3.  The  difference  between  the  active  ampere-turns  and  the  second- 

ary ampere-turns  produced  by  the  direct  current.  These 
two  kinds  of  ampere-turns  tend  to  produce  fields  which  are 
diametrically  opposed  to  each  other  and  which  make  a  phase- 

angle  of  —  with  respect  to  the  exciting  field. 

Practically,  the  theory  is  greatly  simplified  by  the  fact  that  the 
active  ampere-turns  and  the  direct-current  secondary  ampere-turns 
are  substantially  equal  to  each  other  at  all  loads. 

On  the  one  hand,  the  winding  being  a  direct-current  winding,  the 
average  number  of  active  ampere-turns  will  be  given  by  the  formula, 


(a) 


in  which  TV  =  number  of  wires; 

Iw=  active  component  of  primary  alternating  current; 
X=  reduction-coefficient  due  to  the  overlapping  of  the  wind- 

ings, for  an  infinitely  large  number  of  slots. 

The  author  has  published  some  average  values  for  this  coefficient. 
It  should,  preferably,  be  determined  from  existing  machines. 

On  the  other  hand,  the  average  number  of  ampere-turns  due  to 
the  direct  current,  /9,  produced  by  the  converter,  is  equal  to  the  mean 
value, 


1  N/l2\ 

2  2(2)- 


Dividing  (ft)  by  (a)  we  have  the  ratio  of  the  secondary  and  primary 
ampere-turns: 


Whatever  may  be  the  number  of  phases,  q,  if  the  efficiency  be 
assumed  equal  to  unity,   which  makes  the  primary  and  secondary 


GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS    175 

watts  equal,   the  induced  E.M.F.,   as  a  function  of  the   reduction- 
coefficient  k,  due  to  overlapping,  may  be  expressed  thus: 


(  } 


in  which  T—  the  periodic  time, 
and  $=the  magnetic  flux. 

From  this,  we  have 

/2  Ttk 

I  w      V 2 

Substituting  in  (c)  we  have 


I    have   shown  elsewhere    (see  Properties  of  Revolving  Magnetic 

k 

Fields,  Eclairage   Electrique,  1895)  that  the   ratio  —  is  substantially 

K 


fa\2 

equal  to  1  —  1  .     We  therefore  have 


t='^  -     ='1—^^=0.969. 
8\2/      32        32 

This  close  approach  of  the  value  of  £  to  unity  shows  that  the  trans- 
verse reaction  is  practically  negligible  in  comparison  with  the  direct 
reaction.       Hence     rotary 
converters  may  be  treated, 
practically,  as  if  there  were 
no  transverse   reaction  in 
them.  £ 

K,  .  I-.  M.m.f  of  Watfi'sss 

the     internal      lOSSeS  Current  Reaction 

are  taken  into  account,  the 

secondary    (output)    watts 

will  be  from  10  to  15  per 

cent  less  than  the  primary  FlG 

(input)    watts;     and    the 

ratio  ^  will  be  still  less  near  unity.     Nevertheless  it  is  found  that: 

i.  The  direct-current  E.M.F.  undergoes  no  modification  in  con- 
sequence of  the  transverse  reaction,  because  the  brushes  always  remain 


of^  exc/faf/i 


176    GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

absolutely  at  the  neutral  point,  and,  consequently,  only  the  direct 
flux  is  cut  by  the  armature  conductors; 

2.  The  alternating  current  E.M.F.  is  affected  only  to  a  slight 
extent,  because  the  field  resulting  from  the  active  current  Iw,  and 
from  a  direct  current  1 2,  which  is  10  to  15  per  cent  weaker,  consti- 
tutes only  10  to  15  per  cent  of  the  transverse  reaction  field  of  the 
direct-current  armature,  which  field,  as  a  rule,  would  not  exceed  half 
the  direct  field.  But,  even  assuming  that  this  resultant  transverse 
field  may  amount  to  20  per  cent  of  the  direct  field,  the  alternating 
E.M.F.  would  be  thereby  increased  only  in  the  ratio, 

Vl+(o.20)2 

—  =1.02, 

or,  only  about  two  per  cent,  which  is  negligible.  We  are  therefore 
warranted  in  treating  the  rotary  converter  as  a  synchronous  motor 
having  no  transverse  reaction,  as  has  been  done  by  other  authors. 
There  might  possibly  be  an  exception  in  the  case  of  rotary  converters 
having  no  field-excitation;  but  these  are  used  very  little,  and  their 
theory,  in  any  case,  can  be  established  much  more  easily  by  con- 
sidering them  as  transformers. 

As  in  the  case  of  every  alternator,  the  armature,  in  a  rotary  con- 
verter, is  subject  to  magnetic  leakage,  that  is  to  say  a  portion  of  the 
magnetic  flux  produced  by  the  alternating  currents  in  the  armature 
winding  is  shunted  by  the  air  or  by  the  armature-teeth.  This  leakage 
is  offset  in  part  by  the  leakage  which  the  secondary  current  tends  to 
produce,  in  the  contrary  direction  (and  which  offsets,  especially,  the 
effect  of  the  active  current);  hence,  the  resultant  leakage  is  much 
smaller  than  in  a  synchronous  motor.  Nevertheless,  in  order  to 
make  the  theory  general,  we  will  suppose  that  this  magnetic  leakage 
exists;  and  it  will  be  represented,  as  usual,  by  an  inductance,  s,  giving 
rise  to  a  reactance,  cos. 

Factors  Determining  the  Practical  Conditions  of  Operations. 
It  is  very  important  to  note  that  the  operation  of  rotary  converters 
does  not  depend  on  the  characteristics  of  the  machine  only,  but  also 
depends  on  the  constants  of  the  electric  supply-circuit.  It  is,  in 
fact,  the  self-induction  of  the  supply-circuit  and  the  difference  between 
its  E.M.F.  and  the  E.M.F.  induced  in  the  converter,  which  govern 
the  ratio  of  reactive  to  active  currents. 

Every  active  current  lagging  behind  the  E.M.F.  produces  a 
demagnetizing  effect,  whereas  every  reactive  current  leading  the 


GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS    177 

E.M.F.  produces  a  magnetizing  effect.  The  variations  in  the  condi- 
tions of  the  external  circuit,  therefore,  constitute  an  indirect  process 
for  making  the  field-excitation  of  the  rotary  vary,  though  this  variation 
is  often  contrary  to  what  is  wanted. 


I.   CONDITIONS    OF    ELECTRIC-CURRENT    SUPPLY    TO    ROTARY 

CONVERTERS 

From  what  precedes,  it  is  important  to  know,  quite  precisely  and 
quantitatively,  the  manner  in  which  the  reactive  current  varies  with 
the  conditions  of  the  current-supply  to  the  converters,  especially  with 


the  constants  of  the  circuit  and  with  the  impressed  E.M.F.  This 
question  will  now  be  discussed. 

Fundamental  Diagram.  We  can  start  from  the  usual  diagrams 
for  alternators,  made  with  two  scales,  one  for  currents,  and  the  other 
for  E.M.F.'s. 

Assuming  E2=  the  secondary  (D,C.)  voltage  at  the  terminals,  let 
e=OB  (in  Fig.  2)  represent  the  primary  voltage,  which  is  substantially 
proportional  to  £2,  owing  to  the  low  internal  resistance  of  the  arma- 
ture. 

Let  E=OC=the  E.M.F.  of  the  alternator  supplying  current  to 
the  converter  (or  the  voltage  of  the  distribution-system,  according  to 
the  case).  Unless  otherwise  stated  OC  will  be  assumed  constant  at 
aU  loads. 


178   GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

Let  R=  resistance  of  primary  circuit  of  converter; 
X=oj(L+l-\- s)  =  reactance  of  same; 

Z=VR2  +X2  =  impedance  of  same; 

jY" 

tan  r=—=  reactance-coefficient; 
K 

aj=—=  frequency. 

The  ".primary"  circuit  in  this  case  includes  the  armature.  The 
reactance  constant  X  must  therefore  include  the  following:  The 
reactance  (coL)  of  the  reactance-coils  connected  in  series  with  the 
converter;  also  the  reactance  (col)  of  the  alternator  and  of  the  line 
if  the  current  is  supplied  directly  by  an  alternator,  and  not  by  a 
distribution-system;  and,  finally,  the  reactance  (ojs)  corresponding  to 
the  effect  of  the  magnetic  leakage  at  the  armature  itself. 

For  each  value,  6,  of  the  phase-angle  between  the  E.M.F.'s  E 
and  e,  the  third  side,  EC,  of  the  triangle  COB,  represents  the  resultant 
E.M.F.  of  E  and  e,  which  is  equal  to  the  product  of  the  impedance  Z 
by  the  primary  alternating  current  /  corresponding  to  and  'passing 
through  this  impedance.  Constructing  the  triangle  CBH,  on  BC  as 
the  hypothenuse,  with  the  angle  f  at  B  we  will  have 

BH=XI      and      HC=RI. 

In  the  diagrams  <£  will  always  represent  the  angle  of  lag  of  the  current 
behind  the  voltage  at  the  terminals. 

In  order  that  the  vector  BC  may  measure  the  current  /  itself, 
both  in  magnitude  and  in  phase,  it  is  only  necessary  to  take,  for  the 
amperes,  a  scale  which  is  Z  times  greater  than  for  the  volts,  and  to 
take,  as  the  origin  of  phase-angles  (<£),  a  reference  line  BDY,  making, 
with  the  line  OX,  the  angle  f  denned  hereinabove.  The  angle  DEC 
then  measures  a  phase-angle  ($),  representing  a  "  lag  "  if  it  be  to 
the  right  of  ED,  and  a  "lead  "  if  it  be  to  the  left.  We  therefore  count 
the  reactive  current,  /^,  as  positive  on  the  right-hand  side  of  D  and  as 
negative  on  the  left-hand  side. 

If,  from  C  (Fig.  3)  a  perpendicular  CD  be  drawn  to  the  line  of 
reference  BY,  the  segments  CD  and  BD  will  measure  the  reactive  and 
active  currents  for  each  position  of  C.  The  active  current  Iw  can, 
itself,  be  considered  as  equal  to  the  sum  of  two  parts.  One  of  these 
parts  (/o),  represented  by  the  segment  Bd,  is  practically  constant  and 


GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS    179 

corresponds  to  the  approximately  constant  loss  produced  by  the 
"  no-load "  active  current,  in  consequence  of  friction,  resistance, 
hysteresis,  and  eddy  currents.  The  part  /o  will  be  called  the  "  active 
current  for  zero-load."  The  other  part,  represented  by  Dd,  corre- 
sponds to  the  energy  of  the  useful  secondary  current,  and  which  is 
proportional  to  the  said  current. 


Fundamental  Equation.  The  fundamental  equation  connecting 
e,  E,  and  the  active  and  reactive  currents  can  be  written  directly 
from  Fig.  2.  The  triangle  OBC  gives,  in  fact, 


=  e2  +  (ZI)2-2s(ZI)  cos 


In  this  case  we  have 


—  (Z7)  cos  ^=  projection  of  BC  on  OX 

=  (proj.  of  BD  -f  proj.  of  DC)  on  OX 
=  (proj.  of  CH+  proj.  of  BH)  on  OX 
=  ZIW  cos  f+ZId  cos0 
=  ZIW  cos  f+ZId  sin  -jr. 


Substituting  this  value,  and  remembering  that  /2=/t02+/d2,  we 
have,  from  (a), 


.     .     .     (i) 


180    GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

We  also  find  (from  the  triangle  CBH,  in  Fig.  3), 

Z  cos  7-=  R=  resistance, 
Z  sin  f=X=  reactance. 

Substituting  these  values  in  the  last  term,  in  (i),  we  have 


This  equation  enables  the  reactive  current,  Id,  to  be  calculated, 
as  a  function  of  the  load,  i.e.,  of  the  active  current  Iw;  because,  on 
solving  with  respect  to  Z/^,  we  have 


sn 


w  cos  r=o. 


Solving  this  quadratic  equation  for  Id,  we  have 

z[ 


~e  sn 


2  sin2  r+£2-z2/"-2-2£Z/  cos  r  .    . 


This  expression  will  be  put  into  another  form  later. 

Application  of  the  Diagram.  Representation  of  Converter 
Operation  with  Constant  Potential  at  Primary  Terminals  and  at 
Brushes.  If  all  that  is  desired  is  to  maintain  the  potential  constant 
at  the  terminals  of  the  con- 
verter, the  segment  OB=e 
remains  constant,  and,  in 
order  to  predetermine  the 
different  conditions  of 
operation  it  is  sufficient 
to  let  the  point  C,  which 
defines  the  conditions  of 
operation,  describe  a  circle 
around  O  as  a  center, 
with  the  constant  radius 
OC=E.  For  each  posi- 
tion of  C  the  load  is  FIG.  4. 

measured  by  dD  and   the 

reactive  current  by  CD.  It  is  thus  seen,  in  Fig.  4,  that  the  reactive 
current,  whicn  is  positive  for  light  loads  (since  the  characteristic 
point  C  is  to  the  right  of  the  line  BY),  diminishes  as  the  load 
increases  from  the  maximum  value  F  (secondary  load  alone)  to  zero, 
at  G;  and  it  then  changes  in  sign,  and  begins  again  to  increase. 


GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS    181 

If  the  supply  E.M.F.,  E,  is  varied,  which  amounts  to  the  same 
thing  as  changing  the  radius  OC  of  the  circle,  the  particular  load  for 
which  the  reactive  current  vanishes  may  be  varied  at  will.  For 
example,  Fig.  5  (in  which  four  circles  of  different  radii  are  drawn), 
shows  that  it  is  possible  to  have  the  current  in  phase  (power-factor 
equal  to  unity)  with  a  certain  useful  load  dC2  or  with  zero  load,  instead 
of  the  load  dG,  and  that  it  is  possible  even  to  have  negative  lag  (or 
a  leading  current)  at  all  loads,  by  suitable  variation  of  the  supply 
E.M.F.  E. 

It  is  seen  that  in  any  case  it  is  impossible  to  prevent  the  reactive 
current  from  varying  when  the  load  changes.  Since  this  current 
produces  a  magnetizing  or  a  demagnetizing  action  equal  to  that  of 

TV 
K  — Id  V2  inductive  ampere-turns, 


FIG.  5. 

it  is  not  possible  to  maintain  constant  potential  by  means  of  constant 
excitation  if  the  impedance  of  the  circuit  is  not  negligible,  but  that 
it  will  be  necessary  to  increase  or  decrease  the  excitation  ampere- 
turns  by  an  amount  equal  and  contrary  to  that  of  the  ampere-turns 
of  the  reactive  current  in  order  that  the  total  inducing  flux  may 
remain  constant.  (The  variation  of  the  magnetic  leakage  may  intro- 
duce a  slight  complication,  as  will  be  seen  later.) 

With  a  given  value  of  /-,  if  the  impedance  Z  is  varied,  we  see  that, 
the  greater  its  value,  the  larger  the  scale  of  amperes  will  be,  and,  con- 
sequently, the  smaller  will  be  the  variation  of  load  which  corresponds 
to  a  given  angular  displacement  of  the  vector  OC.  The  variations 
of  the  active  current  with  the  load  will  therefore  increase  in  propor- 
tion with  the  impedance. 


182    GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

If  the  impedance  is  varied  by  changing  only  the  reactance,  X, 
without  changing  the  resistance,  R,  the  result  is  less  clearly  seen, 
because  the  direction  of  the  reference  line  OY  then  changes  at  the 
same  time  as  the  scale  of  amperes.  This  point  will  be  discussed  later 
under  the  general  case. 

General  Case.  Reactive  Current  Values  for  a  Given  Voltage 
Variation  as  a  Function  of  the  Load.  In  the  most  frequent  case  the 
voltage  is  raised  slightly  when  the  load  increases,  to  compensate  for 
the  line-losses. 

In  Fig.  6,  let  £=the  E.M.F.  corresponding  to  zero-load,  and  let 
E',  E",  E'",  etc.,  represent  the  successive  values  which  E  must  have 
when  the  load  corresponds  to  the  active  currents  Tw',  Iw",  Iw"', 


etc.  These  current-values  are  to  be  measured  off  on  a  reference-line 
BY,  starting  from  the  point  B,  giving  the  distances  BD',  BD",  ED'", 
etc.,  and  the  E.M.F.'s  are  to  be  measured  off,  along  the  horizontal 
line  BO  starting  from  B  toward  the  left,  giving  the  distances  BO, 
BO',  BO",  BO'",  etc. 

From  each  of  the  "  O  "  points  thus  determined,  as  a  center,  let 
a  circular  arc  be  drawn,  with  the  same  radius,  equal  to  the  external 
E.M.F.  E,  which  is  a  constant,  assumed  to  be  known.  (In  Fig.  6, 
in  order  to  make  the  case  more  general,  it  is  assumed  that  the  E.M.F.'s 
may  be  unequal.)  The  respective  points  of  intersection  of  the  circular 
arcs  with  the  lines  dC,  D'C',  D"C",  D"'C"',  drawn  perpendicular  to 
the  reference-line  BY,  give  the  conditions  corresponding  to  the  various 
loads. 


GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS    183 

The  reactive  currents  corresponding  to  these  loads  are  therefore 
entirely  determined. 

It  will  be  seen  that  there  is  an  infinite  number  of  possible  solutions, 
according  to  the  value  taken  for  the  constant  E.M.F.  E.  This  E.M.F. 
can  be  made  high  or  low,  as  desired.  It  depends  on  the  ratio  of  trans- 
formation of  the  static  transformers  used  for  supplying  current  to 
the  converters. 

Moreover,  the  impedance  Z  can  also  be  regulated,  within  certain 
limits,  at  least,  by  the  introduction  of  reactance  in  the  current  supply- 
circuit.  (It  is  obviously  desirable  to  reduce  the  ohmic  resistance  to 
the  lowest  possible  value,  to  avoid  lowering  the  efficiency. )  It  remains 
to  be  seen  what  are  the  most  suitable  values  for  E  and  X.  This  point 
will  be  discussed  in  the  next  two  sections. 

Most  Suitable  Value  of  Current-Supply  Voltage.  This  value  is 
evidently  that  which  gives  zero-lag  (unity-power-factor)  for  the  most 
usual  or  frequent  load,  because  we  then  have,  for  that  load,  a  minimum 
current-value,  consequently  better  line-efficiency,  and  better  "  over-all  " 
efficiency.  This  condition  of  operation  can  be  determined  from  the 
known  or  expected  conditions  of  service,  for  the  particular  apparatus. 

One  might  be  tempted,  on  general  principles,  to  take  the  maximum 
load  as  the  one  for  which  the  supply-voltage  is  to  be  specially  adjusted, 
so  as  to  give  the  best  results  with  that  load,  because  it  is  precisely 
then  that  it  is  most  important  to  reduce  the  losses;  but  since,  in 
practice,  it  is  necessary  to  consider  the  mean  daily  heating  and  effi- 
ciency of  the  apparatus  used,  it  is  for  the  average  or  normal  load,  and 
not  for  excessive  loads  of  momentary  duration  only,  that  the  supply- 
voltage  should  be  adjusted,  with  the  object  of  obtaining  unity  power- 
factor. 

The  range  of  voltage-regulation  desired  in  the  rotary  converter 
can  be  obtained  with  more  ease  and  precision  when  the  point  of 
maximum  power-factor  ("  no  lag  ")  corresponds  to  the  average  load. 
In  fact,  as  will  be  seen  later,  the  only  converter- voltages  (D.C.)  which 
can  be  obtained  exactly,  by  practical  methods  of  excitation,  are  those 
corresponding  to  zero-lag  and  zero-load. 

Most  Suitable  Value  of  Reactance.  It  is  important  to  note,  at 
the  outset,  that  a  certain  minimum  reactance  is  necessary  in  order 
to  obtain,  with  a  constant  (A.C.)  E.M.F.  E,  a  (D.C.)  voltage  which 
will  increase,  or  even  remain  constant,  with  the  load. 

Let  us  suppose  that  there  is  no  reactance  in  the  circuit.  This 
makes  f=o;  the  reference-line  BY  (Fig.  7),  will  then  coincide  with 


184    GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 


the  axis  OX;  and  the  currents  corresponding  to  increasing  loads  will 
be  represented  by  the  distances  Bd,  BD' ,  BD",  etc.  Let  D",  for 
example,  represent  the  normal  load  condition  for  which  there  is  no 
lag  (?-=°).  The  vectors  Cd,  C'D',  C"D",  which  measure  the  corre- 
sponding reactive  currents,  increase  extremely  fast  at  light  loads; 
and,  since  the  scale  of  current-values  is  very  small,  owing  to  the  fact 
that  Z  is  very  small,  it  is  seen  that  the  reactive  currents  at  light  loads 
would  be  very  large.  On  the  other  hand,  in  the  case  represented 
by  Fig.  6,  owing  to  the  fact  that  the  load-lines  are  no  longer  perpen- 
dicular to  OB,  it  is  easy  to  find,  with  a  constant  E.M.F.  E,  and  with 
small  active  currents,  load-points,  C,  C',  C",  corresponding  to  points 

0"     O'     O  €  Bat  A'  \D*     Y 


FIG.  7. 
O,  O',  O",  which  are  displaced  to  the  left,  and,  consequently  cause 

/£\ 

an  increase  in  the  voltage  I  j-  )  at  the  brushes. 

It  is  seen,  thus,  that  it  is  possible  to  obtain  economically  a  voltage- 
regulation  giving  constant  or  rising  voltage,  as  the  load  increases, 
only  when  the  circuit  contains  a  certain  amount  of  reactance. 

This  point  being  established,  it  is  easy  to  note  the  influence  of 
reactance,  by  supposing  that  the  "  no  lag "  condition  is  made  to 
correspond  always  to  the  same  load,  i.e.,  to  the  same  value  of  the 
active  current,  /„,.  In  fact  (Fig.  8),  all  the  load-points  corresponding 
to  that  condition  are  found  to  be  located  at  distances,  ZIW,  from  the 
point  B,  which  are  proportional  to  the  impedances  Z=\//R2—X2; 
hence,  if  we  measure  off,  on  OB  prolonged,  a  distance  BK,  equal 
to  RIW,  all  the  points  G,  corresponding  to  different  reactances  X, 
will  be  located  on  the  right  line  KP  perpendicular  to  OK.  Likewise, 


GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS    185 


the  point  of  zero-load,  d,  will  be  on  the  perpendicular  kd  corresponding 
to  a  segment  Bk=RjQ.  We  will  compare,  for  various  cases,  the  values 
of  the  reactive  current  iQ  corresponding  to  this  no-load  condition. 

This  reactive  current  is  obtained  by  drawing  through  d  the  right 
line  dF,  perpendicular  to  BG,  until  it  crosses  the  circle  of  radius  E. 
In  order  to  make  this  cur- 
rent-value comparable  with 
BK,  it   is  only    necessary 
to  reduce  also  the  segment 

D 

dF   in   the    proportion  — , 

£i 

which  can  be  done  by  pro- 
jecting it  on  KG.  The 
segment  d'F'  then  meas- 
ures the  value  Rio,  while 
Bk  measures  RIW.  By 
this  process  all  the  current  FlG  g 

values  are  referred  to  one 

and  the  same  scale,  IQ,  whatever  may  be  the  value  of  Z.  On  making 
the  diagram  anew  for  cases  corresponding  to  various  angles  of  the  line 
of  reference  BG,  it  is  found  that  the  reactive  current,  which  is  very 

large  for  very  small  "  lags  " 
diminishes,  rapidly  at  first, 
then  more  slowly. 

In  order  to  follow  more 
"~  K  easily  the  law  of  variation, 
let  us  suppose  the  active 
current  to  vanish  at  zero- 
load,  i.e.,  let  us  assume 
jo=o,  which  makes  d  coin- 
cident with  D  (Fig.  10). 
The  reactive  current  IQ  is 
then  proportional  simply 
to  the  distance  FH. 
If  the  point  G  be  raised  gradually  on  the  vertical  line  KG,  starting 

T>  T? 

from  the  point  K,  the  reactive  current  is  at  first  equal  to  — ~  (Fig.  9), 

J\ 

and  it  will  then  decrease;  for  example,  for  the  point  G  (Fig.  10),  it 
will  decrease  to  FH.  When  G  is  very  high,  it  is  seen  that  an  addi- 
tional rise,  G'G",  for  example,  will  increase  almost  proportionally  the 


FIG.  9. 


186    GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

radius  E  without  altering  materially  the  inclination  of  the  segment 
BF'.  Hence,  if  the  reactance  is  increased  beyond  a  certain  point 
the  reactive  current  ?o,  corresponding  to  zero-load,  will  increase  with  it. 
This  reactive  current  must  therefore  have  a  minimum;  and  to 
every  value  higher  than  this  minimum  there  will  correspond  two 

reactance  values,  and  two  inclina- 
tions of  the  line  BG,  one  greater, 
the  other  smaller. 

We  can  learn  more  about  this 
minimum  by  calculations  based  on 
the  fact  that  the  same  value  of 
the  supply-voltage  E  enables  both 
the  "  no-lag  '  point  G  and  the 
"  no-load  "  point  F  to  be  obtained. 

To  make  the  case  more  general,  let  us  suppose  a  condition  wherein 
/o  does  not  vanish  and  wherein  the  voltage  with  load,  e',  is  different 
from  the  voltage  s  without  load.  We  will  have  the  two  following 
relations: 


FIG.  10. 


At  the  point  G,     E2=  e'2+Z2Iw2  +  2e'RIw. 


At  the  point  F,     E2 
From  these,  eliminating  E,  we  have 


(«) 
<*) 


(4) 


This  is  the  condition  required.      Solving  for  iQ  and  then  for  X, 
we  obtain  the  two  following  formulas: 


Z2 


u,2  -if)  +  2  e'RIw] 


(V) 


In  (5)  the  expression  under  the  radical  sign  is  taken  with  the  +  sign 
only,  because,  for  each  value  of  X,  i$  can  have  only  one  value,  which, 
necessarily,  is  positive.  On  the  other  hand  it  has  been  seen  that  X 
has  two  values  for  each  value  of  «0;  hence  it  is  necessary  to  retain 
both  signs  (-[-)  before  the  expression  under  the  radical  sign,  in  Eq.  (6). 


GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS    187 

The  minimum  value  of  IQ  could  be  obtained  by  making  the  derivative 
of  Eq.  (5)  equal  to  zero  and  solving;  but  it  is  simpler  to  equate  to 
zero  the  expression  under  the  radical  sign  in  Eq.  (6),  because  the 
minimum  value  of  i0  is  evidently  that  below  which  the  value  of  X 
would  become  imaginary.  Hence,  the  positive  and  negative  quantities 
under  the  radical  sign  must  offset  each  other,  when  there  is  a  minimum. 


Equating  the  quantity  under  the  radical  sign  in  (6)  to  zero,  and 
simplifying  with  respect  to  (/w2—  io2),  instead  of  *o2,  we  have 


Since  the  ohmic  drop  of  voltage  should  be  only  a  small  fraction  of  E, 
the  last  term,  R2(Iw2i0)  may  be  neglected,  and  we  thus  obtain  the 
approximate  value 


and  the  minimum  value  of  io2  is 


P'2_ff2  i  -,-'/ 

C         C       T  ^  c    •** 

-   g,2+ 


Since  the  quantity  under  the  radical  sign  in  (6)  practically  vanishes 
for  this  minimum  value  of  i02,  the  expression  for  X  which  corresponds 
to  the  minimum  reactive  current  IQ  is,  simply,  by  approximation, 


188    GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 
or,  substituting  for  iQ, 


This  is  the  most  suitable  reactance  -value  for  reducing  iQ  to  a  min- 
imum. Inasmuch  as,  on  the  other  hand,  the  supply-voltage  required 
increases  with  the  reactance  X,  and  inasmuch  as  this  voltage  must 
be  kept  as  low  as  possible  in  order  to  utilize  the  apparatus  to  the  greatest 
advantage  (i.e.,  to  obviate  the  necessity  of  making  it  too  large  and, 
consequently,  more  expensive),  for  the  effective  output,  we  may  con- 
clude that,  in  general,  the  reactance  value  should  be  comprised  between 
zero  and  the  value  (Xo)  corresponding  to  the  minimum  zero-load 
reactive  current. 

NOTE  i.  —  Formulas  5,  6,  7,  8  can  be  put  into  a  form  which  is  often 
more  convenient  practically,  by  expressing  the  values  of   /o  m  terms 
of  Iw,  the  values  of  RI  in  terms  of  e,  the  values  of  c'  in  terms  of  s, 
and  the  values  of  X  in  terms  of  R. 
Let  us  take 

e'  RIW  X  I 


e 


~ 

K  1 


w 


cos  (j)  being  the  power-factor,  and  m  being  the  value  of  tan  7-. 
From  these  equations,  we  deduce 


XIW 

=am, 


Z    a\/i+m2' 
and  also  the  equivalent  formulas: 


p  __          ___  _ 

a(i-tah2^)±    \a2(i-tan2^)         i-tan2f 

.    .  '    .  .o       T    9  (l+£)2  —  I+2a'(l+£)  /n 

minimum  of  V^  S-J—-J,    .     .: (;') 

].    •     .     (8') 


GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS    189 

In  the  particular  case  where  the  voltage  must  be   constant,  we  must 
have  e'=  s,  or  e=o,  and  the  expression  for  iQ2  reduces  to  the  simple  form 


«?=IJ-2- 


which  gives,  substantially, 


X0=—\/2a(i  +  2a), 


or,  practically,  X0=R    I— . 

NOTE  2. — In  the  particular  case  where  iQ=Iw  and  tan  <£=i,  for- 
mulas (7)  and  (7')  will  no  longer  apply,  and  they  must  be  replaced  by 
the  following  formulas,  deduced  from  Eq.  (4) : 


*  r(l+£)_f!±££l ,   (7a, 

tan^L  2a 


Regulation  of  Voltage  at  Terminals  by  Variation  of  the  Supply 
E.M.F.  In  what  precedes  it  has  been  supposed  that  the  E.M.F.  E 
of  the  source  of  current  is  constant  at  all  loads.  This  necessitates  a 
reactive  current  at  light  loads. 

It  is  possible  to  compound  or  to  regulate  automatically  the  excita- 
tion of  the  alternator  in  such  manner  that  its  E.M.F.,  E,  will  increase 
with  the  load.  This  compounding  will  obviate  all  necessity  for  any 
reactance;  and  the  line  of  reference  in  the  diagram  can  be  the  line 
OB  prolonged,  if  there  is  no  other  loss  of  voltage  than  that  due  to 
ohmic  drop.  This  solution  is  obviously  the  most  perfect  one,  theoret- 
ically, but  it  cannot  be  realized  exactly  otherwise  than  in  the  excep- 
tional case  where  the  rotary  converters  are  supplied  individually  by 
separate  alternators.  It  is  not  possible  otherwise  to  establish  a 
correlation  between  the  load  of  the  converters  and  that  of  the  lines 
when  they  supply  current  at  the  same  time  to  other  apparatus. 

An  approximation  to  this  method  of  regulation  can  be  made  by 
supplying  the  converters  with  current  through  transformers  whose 


190   GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

ratio  of  transformation  is  made  variable  by  means  of  "  taps  "  on  the 
the  secondary  winding  connected  with  suitable  switching  apparatus. 
The  fundamental  diagram  (Fig.  12),  or  the  formula  (a),  will  deter- 
mine the  values  which  should  be  given  to  E  for  each  value  of  the  E.M.F. 
e,  and  of  the  load  ZIW,  measured  off  along  the  line  BY  corresponding 
to  the  fixed  reactance  of  the  circuit.  The  most  suitable  value  for  the 
reactance  is,  in  this  case,  evidently,  the  lowest  value,  X=o,  when 
possible;  because  this  value  reduces  to  a  minimum  the  values  of  the 
E.M.F.  E  required.  Therefore,  no  supplementary  reactance  should 
be  included  in  the  circuit.  Under  these  circumstances  BY  becomes 
the  prolongation  of  OB  in  the  diagram. 


FIG.  12. 

Regulation  of  Voltage  at  Terminals  by  Variation  of  Reactance. 

It  is  also  possible  to  neutralize  the  lag  at  all  loads  more  simply  by  keep- 
ing the  supply  E.M.F.  E  constant  (and  equal  to  that  required  to 
insure  the  necessary  voltage  at  the  terminals  with  full  load  without 
reactive  current),  and  then  reduce  this  voltage  at  lighter  loads  by  the 
introduction  of  suitable  reactances  in  the  circuit. 

Fig.  13  explains  this  method  of  regulation  in  the  case  where  the 
E.M.F.  E  is  constant;  and  it  enables  the  characteristic  features  of  this 
case  to  be  studied. 

Let  BK  =  RIW ,  represent  the  loss  of  voltage  by  ohmic  drop  in  the 
circuit,  with  full  load.  The  segment  BK  can  serve  to  represent  the 
acti  e  current. 

Let  BK/  BK,"  etc.,  represent  intermediate  current  values;  and 
let  the  corresponding  perpendiculars  KG,  K'G/  K'lG"  be  drawn. 
Their  points  of  intersection  G,  G/  G",  with  the  circle  of  radius  E, 


GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS    191 

described  around  the  center  O,  will  define  the  load-conditions  correspond- 
ing to  these  currents. 

In  order  to  obviate  reactive  currents  it  is  sufficient,  for  each  case, 
that  the  reactance  should  be  adjusted  to  a  value  exactly  equal  to  that 


FIG.  13. 

which  makes  the  reference  line  pass  through  G.     This  means  that  the 
reactance  X  must  be  made  successively  equal  to  the  following  values: 


BK    '     BK 


R-  R.  etc 

'     '     BK"     '       * 


These  values  of  X  can  be  calculated  for  each  value  of  Iw,  by  the   Eq. 
(a)  given  herein  above.     From  this  equation 


(a) 


on  substituting  for  Z2  its  value  (Z2=R2+X2),  and  solving  for  X2,  we 
have 


X2=-R2  + 


E2-e'2-2e'RIw 

Iw2 


Practically,  this  regulation  can  be  worked  by  hand  or  even  auto- 
matically, by  making  use  of  reactance-coils  having  movable  cores, 
which  are  pushed  all  the  way  in  when  the  load  is  zero,  and  are  gradually 


192   GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

pulled  out  as  the  load  increases,  in  such  a  way  as  to  bring  back  to  zero 
a  direct-reading  phase-indicator,  connected  at  the  terminals  of  the 
converters. 

This  method  is  applicable  whatever  may  be  the  value  of  E  (i.e., 
the  radius  of  the  circle);  but  the  most  suitable  value  is,  obviously, 
the  smallest  value,  which  corresponds  to  E=sf+RI. 

The  diagram  presupposes  that  the  locus  of  the  point  G  is  a  circle; 
but  if  the  E.M.F.  e  varies  the  center  O  of  that  circle  is  displaced  and 
the  locus  of  G  is  a  curve,  which  can  be  drawn  by  points  determined 
from  the  data.  This  does  not,  in  any  way,  necessitate  modifications 
in  the  method. 

Power-Factor  of  the  Generator.  In  what  precedes,  we  have  seen 
the  necessity  of  keeping  down  the  value  of  the  E.M.F.,  e.  It  is  cus- 
tomary to  express  the  disadvantages  of  the  useless  increase  of  this 
E.M.F.  by  stating  the  value  of  the  power-factor,  i.e.,  by  the  value  of  the 
cosine  of  the  phase-angle,  usually  termed  the  "  angle  of  lag,"  between 
the  E.M.F.  and  the  current. 

Let  </>  denote  this  phase-angle,  and  let  0  denote  the  phase-angle 
between  e  and  E;  and  let,  as  before,  (f>  denote  the  phase-angle  between 
the  current  and  the  E.M.F.  e.  We  have 


tan  </>=-*, 

*  w 

and  this  angle  of  lag,  c£,  is  always  in  the  same  direction  as  6  (see  Fig. 
i),  i.e.,  in  the  direction  of  the  hands  of  a  watch  (in  these  diagrams). 
We  therefore  have 


For  the  condition  of  average  load,  that  is  to  say,  in  the  cases  where 
there  is  no  lag,  we  have  </>=o,  and,  therefore,  '</>  reduces  to  the  same 
value  as  6  itself. 

In  order  to  increase  the  power-factor  of  the  generator  as  much  as 
possible  at  average  loads,  we  should,  therefore,  reduce  0,  and,  conse- 
quently, e,  as  much  as  possible,  as  already  stated.  It  is  much  more 
important  to  do  this  than  to  reduce  to  zero  the  "  no-load  "  lag  between 
the  current  and  the  converter  E.M.F.  Therefore,  the  reactance  used 
should  have  a  value  comprised  between  zero  and  the  value  which 
renders  the  no-load  lag  a  minimum.  This  is  all  the  more  desirable 


GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS    193 

because  the  ]ag  6  then  becomes  negative  at  light  loads,  as  indicated 
in  Fig.  14,  and  the  value  of  </>  is  then  a  difference  (p=<j>  —  6f,  which 
increases  but  slowly  with  <£. 


FIG.  14. 

It  is  logical  to  endeavor  to  make  6  and  <£  —  6'  equal  to  each  other, 
in  such  a  way  as  to  render  the  two  extreme  values  of  <f>  as  equal  and  as 
small  as  possible. 


CHAPTER    II 

PREDETERMINATION  OF  THE  FIELD-EXCITATION  OF  ROTARY 

CONVERTERS 

IN  the  preceding  chapter  we  discussed  the  conditions  attending 
the  supply  of  electric  current  to  rotary  converters,  and,  in  particular, 
the  effects  of  reactance  in  the  supply-circuit,  and  the  various  ways 
of  using  it  to  obtain  a  given  range  of  impressed  voltage  for  a  given 
range  of  load  without  exceeding  a  certain  maximum  reactive  current, 
and  even  while  making  this  current  approach  a  minimum  value.  There 
still  remains  the  second  portion  of  the  problem  to  be  solved,  namely, 
the  determination  of  the  field-exciting  ampere-turns,  due  to  either  the 
series  or  the  shunt-winding  coils,  which  enable  the  rotary  converter  to 
follow  approximately  this  law  of  variation,  in  consequence  of  the  effects 
of  the  reactive  currents. 

In  what  follows  it  will  be  assumed  that  we  are  dealing  with  the 
ordinary  case  of  a  converter  which  is  operating  without  lag,  at  its  average 
load,  and  which  consequently,  has,  at  lighter  loads,  a  positive  reactive 

current  f  lagging  —  behind  the  E.M.F.  j  whereby  there  is  produced  a 

magnetizing  effect  on  the  field.1  This  is  the  case  which  was  discussed 
in  the  preceding  chapter. 

Characteristic  Features  of  the  Rotary  Converter.  We  will  begin 
by  considering  the  magnetic  features  of  the  rotary  converter,  which 
involve  two  new  important  elements: 

i.  The  excitation-curve,  or  the  curve  showing  the  variation  of  in- 
duced E.M.F.  as  a  function  of  the  field-exciting  ampere-turns  due  to 
the  field-windings.  This  curve  is  supposed  to  be  known  from  the 
shop-tests  made  of  the  machine  at  the  time  it  was  finished. 

In  the  excitation-curve  (Fig.  15)  the  ordinates  represent  the  E.M.F. 's 

1  It  is  easy  to  see  that  any  reactive  current  which  lags  behind  the  impressed 
E.M.F.  EI  is  leading  with  respect  to  the  internal  E  M.F.  e,  since  E  and  E  are 
practically  in  opposition. 

194 


FIELD-EXCITATION  OF  ROTARY  CONVERTERS        195 

(volts)  induced,  and  the  abscissas  represent  the  corresponding  ampere- 
turns  of  excitation. 

2.  The  equivalent  magneto-motive  force  of  the  armature,  i.e.,  the 
number  of  ampere-turns  which,  if  acting  upon  the  field-cores  (same  as 
the  ampere-turns  of  the  regular  field-winding)  would  produce  in  the 
magnetic  circuit  the  same  magnetomotive  force  as  is  produced  by  the 
reactive  current  Id  circulating  through  the  armature. 

These  equivalent  average  ampere-turns  can  be  represented,  as  in 
an  alternator,  by  the  expression, 


in  which  N'=  the  number  of  peripheral  wires  of  the  armature-winding 
per  double  field  (i.e.,  per  magnetic  circuit);  K  being  a  coefficient  of 
reduction  which  depends  upon  the  number  of  phases  and  of  armature 
slot's,  and  on  the  width  of  the  poles;  and  7j  being  the  effective  value 
of  the  reactive  current. 

K  is  determined  by  calculation,  by  taking  /</=!  ampere,  with 
zero-lag,  and  calculating  the  mean  magnetic  potential  produced  under 
one  pole. 

The  value  of  K  can  be  obtained  by  experimental  measurement. 
This  can  be  done  with  an  approximation  which  is  generally  sufficient 
for  all  purposes  (since  the  leakage-reactance,  'ws,  is  practically  negligible), 
by  running  the  converter  at  normal- speed  by  power  applied  mechanically, 
and  then  making  it  supply  (on  the  A.-C.  side)  with  normal  voltage,  a 
purely  inductive  external  circuit  (composed  of  a  reactance  giving  a 
power-factor — cos  <^> — equal,  at  most,  to  0.20);  the  difference  between 


196   GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

the  voltage  (s]  at  the  terminals,  with  this  load,  and  the  voltage  (e) 
obtained  with  open  circuit,  with  the  same  excitation,  is  the  measure, 
in  volts,  of  the  armature-reaction  corresponding  to  the  current  supplied 
to  the  external  circuit  by  the  converter. 

Taking  the  values  e  and  e  thus  obtained  and  referring  them  to  the 
excitation-curve  (Fig.  15),  they  will  correspond  to  different  points  on 
this  curve.  The  difference  between  the  abscissas,  A  and  a,  correspond- 
ing to  these  two  points  will  be  a  measure  of  the  equivalent  ampere- 
turns  of  the  armature  (neglecting,  as  before,  the  effects  due  to  magnetic 
leakage).  It  is  only  necessary,  in  order  to  obtain  K,  to  divide  a  —  A 
by  the  product  of  the  effective  reactive  current  Id  measured  in  the  above- 
mentioned  experimental  test,  by  half  the  number  of  turns  of  wire, 

Nf  /- 

—  ,  and  by  the  constant  V  2.    We  will  have 


Compound-Excitation.     Different    Factors    of    this    Excitation. 

Rotary  converters  are  excited  by  current  taken  from  their  secondary 
(D.-C.)   side.     The  field-excitation  may,  obviously,  be  either  shunt, 
series,  or  compound. 
Let  £2  =  secondary  voltage  at  the  terminals; 

1  2=  secondary  or  output  current  delivered  by  the  rotary  converter; 
n=  number  of  turns  of  the  series-  winding; 
r=  resistance  of  one  turn  of  the  shunt-  winding. 
The    shunt   ampere-turns   and   the    series    ampere-turns   will   be, 
respectively,  equal  to 

-~      and      w/2. 

It  is  these  ampere  turns  which,  conjointly  with  the  equivalent 
ampere-turns  of  the  armature,  determine  the  total  excitation. 

In  the  most  general  case,  that  of  a  compound-wound  converter, 
the  question  is,  therefore,  quite  complex,  since  there  are  three  variable 
factors,  the  first  of  which  varies  with  the  voltage,  the  second  with  the 
output  (direct)  current,  and  the  third  with  the  reactive  (alternating) 
current. 

As  already  seen,  the  secondary  ampere-turns  are  proportional  to 
the  ampere-turns  due  to  the  useful  primary  current,  IW—JQ. 


FIELD-EXCITATION  OF  ROTARY  CONVERTERS         197 

Let  £  represent  the  ratio  of  transformation,  i.e.,  the  ratio  of  the 
primary  E.M.F.  to  the  secondary  E.M.F,  Since  the  armature-resistance 
is  already,  by  hypothesis,  counted  once  in  that  of  the  supply-circuit, 
it  need  not  be  counted  again  here.  We  can,  therefore,  write: 

Ampere-turns  of  shunt  excitation, 


Ampere-turns  of  series-excitation,1 

w-j0)-,     .     .          .     .     (18) 


Ampere-turns  of  reactive  currents,  produced  by  the  armature, 


/KN' 
(-7= 

\V  2 


'\ 
Aa=-7=)Id  .........     ;     (19) 

/ 


The  magnetomotive  forces  are  thus  expressed  directly  in  terms  of  the 
elements  of  the  primary  currents.  This  will  enable  the  excitations  for 
each  load-condition  to  be  calculated. 

If  we  count  as  positive  the  lagging  reactive  currents  (which  are 
magnetizing  currents)  the  total  number,  A,  of  exciting  ampere-turns 
corresponding  to  the  load-condition  elw  will  therefore  be 

e      nkn  .  KN' 


The  -f  sign  is  placed  before  the  reactive  ampere-turns  because,  under 
the  normal  conditions  of  current-supply  previously  considered,  the 
reactive  current  is  directed  to  the  right  of  D,  i.e.,  it  is  positive  and 
magnetizing  between  zero-load  and  normal  load,  and  it  is  demagnetiz- 
ing beyond  normal  load.  The  +sign  is  placed  before  the  series  ampere- 


1  From  the  relation  found  at  the  beginning  of  Chapter  I,  namely,  —  =  —  =,  we 

Iw     V  2 

get  72=  -  —  Iw.     In  Eq.  (18),  for  the  sake  of  greater  precision,  Iw  is  replaced  by 

\/2 

/„,  —  /„.     If  the  armature  had  two  different  windings,  the   ratio  would  have  to 

N 
be  multiplied  by  that  of  the  numbers  of  turns  of  these  two  windings,  —  . 


198   GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

turns  because,  in  general,  the  series-  winding  must  work  concordantly 
(or  cumulatively)  with  the  shunt-winding,  as  will  be  seen  later. 

The  unknown  constants,  r  and  n,  of  the  windings,  are  easily  deter- 
mined when  the  E.M.F.,  £0  for  zero-load,  and  the  E.M.F.  em  for  the 
mean  load  (corresponding  to  the  current  Im,  with  zero-lag)  are  known. 

The  excitation-curve  (Fig.  15)  gives  the  ampere-turns,  AQ,  Amt 
corresponding,  respectively,  to  the  E.M.F.'s  £o  and  £«*•  For  the 
normal  load-condition  we  will  have 


.    N 

(21) 


For  the  no-load  condition  we  will  have 

KN'. 


(22) 

$r        \  2 

As  already  seen,  the  value  of  the  reactive  current  corresponding 
to  zero-load  cannot  fall  below  a  certain  minimum.  If  a  minimum 
value  be  assumed  for  this  current,  Eq.  (22)  can  be  used  to  determine 
the  conductance  of  one  turn  of  the  shunt  winding.  From  Eq.  (22), 

solving  for  —  ,  we  have 


From  Eq.  (21),  solving  for  n,  and  substituting  for  —  the  value  given 
in  Eq.  (23),  we  have 


_  V—  -  _          __  _  /    \ 

Tlk  (Im  —JQ)  Tlk  (Im  —jo) 

It  is  seen  that  the  shunt  ampere-turns,  which  are  proportional  to  —  , 

are  less  than  the  total  ampere-turns  A0  (on  the  assumption  that 
£m  >  £0),  and  that  they  decrease  when  the  reactive  current  i0  increases; 
whereas  the  series  ampere-turns,  which  are  proportional  to  n(Im—jo) 
increase  with  the  reactive  current  i$. 

Determination  of  Reactive  Current  as  a  Function  of  the  Excita- 
tion, when  the  Active  Current  is   Constant,  then  when   the  Power 


FIELD-EXCITATION  OF  ROTARY  CONVERTERS         199 


is  Constant,  the  Generator  E.M.F.  being  always  Constant.     Let  us 

suppose,  as  in  the  case  of  synchronous  motors,  that  the  converter  is 
supplied  by  a  constant  E.M.F.  E,  and  excited  by  an  exciting  current 
which  can  be  regulated  at  will. 

The  fundamental  diagram  (Fig.  2),  combined  with  the  preceding 
curve  (Fig.  16),  will  enable  us  to  find  readily,  for  each  value  of  the 
active  current,  Iw,  the  values  of  reactive  current  corresponding  to 
each  value  of  the  excitation,  expressed  in  ampere  -turns. 

Let,  for  example,  Iw  be   the  given  active  current,  and  BD  the 
line  of  reference,  which  is  drawn  after  the  value  of  7-  has  been  obtained. 
The  locus  of  the  point  C,  characteristic  of  the  load-conditions,  for  that 
constant  value  of   Iw,  is  the 
right    line    DQ,  drawn    per- 
pendicular   to    BD,    at    the 
distance   BD=ZIW,    whereas 
the  locus  of  the  point  O  the 
second  end  of  the  vector  E, 
is  the  line  OBX. 

All  the  load-conditions 
corresponding  to  the  active 
current  Iw  will  therefore  be 
obtained  by  drawing,  from 
the  first  points  C,  C",  C7",  .  .  ., 
on  the  first  right  line,  circular 
arcs  of  radius  E,  intersecting 
the  second  line  at  the  points 

O,  O',  O",  ....  The  segments  BO,  BO',  BO",  .  .  .  thus  formed 
measure  the  values  of  the  corresponding  internal  E.M.F.  's,  £,  e', 
e",  .  .  .,  and  the  segments  DC,  DC',  DC",  .  .  .  divided  by  Z,  measure 
the  values  of  the  reactive  current  7j. 

For  each  pair  of  values  of  e  and  Id,  thus  determined,  the  correspond- 
ing ampere-turns  of  excitation,  Ae,  are  obtained,  from  the  excitation- 
curve  (Fig.  1  6),  by  subtracting  the  counter  ampere-turns  of  the  arma- 
ture from  the  resultant  ampere  turns  (A)  corresponding  to  the  ordinate 
(s)  of  this  curve;  whence  we  have 


. 

It  should  be  remembered  that  the  reactive  current  Id  is  demagnetiz: 
ing  and  positive  if  it  is  counted  to  the  right  of  the  axis  BD,  and  that 


200  GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

it  is  magnetizing  and  negative  if  counted  to  the  left.  The  upper  sign 
applies  in  the  first  case,  and  the  lower  sign  in  the  second  case.  Under 
those  conditions  the  diagram  can  now  be  used  for  determining  the 
locus  of  the  points  C  and  the  variations  of  E,  when  the  power  is  con- 
stant, if  the  generator  E.M.F.  be  maintained  constant  as  in  the  case 
of  synchronous  motors.  We  have  in  this  case,  simply, 

P=elv, 

so  that,  for  each  value  of  Iw,  and,  consequently,  for  each  position  of  the 
right  line  DQ,  the  value  of  e,  and,  therefore,  the  position  of  the  point 
O,  are  known;  and,  from  these  points,  circles  of  radius  E  can  be  drawn, 


/  \  /\c 

' V  \    /'v 

/    \~V  5 


-X 


Yj 

FIG.  17. 


whose  respective  intersections  with  the  right  lines  DQ  give  the  C 
points  required.  It  is  possible,  thus,  to  draw  the  "  V  "  curves  analogous 
to  those  of  synchronous  motors,  and  the  curves  of  the  corresponding 
values  of  e.  But  these  load-conditions  for  constant  power  no  longer 
present  in  the  case  of  rotary  converters  the  same  practical  interest 
as  in  the  case  of  synchronous  motors,  because  the  -variation  of  £  would 
be  very  troublesome. 

Different  Values  of  the  Excitation,  with  Constant  Power  and 
Constant  Potential.  V-Curves  for  Constant  Potential.  The  pre- 
ceding diagram  '(Figs.  16  and  17)  also  gives  the  immediate  solution 
of  the  more  practical  problem  l  which  consists  in  determining  the 
current-values,  7,  corresponding  to  different  excitations,  with  '  constant 

1  The  case  considered  is,  as  yet,  more  of  theoretical  than  of  practical  interest, 
because  it  assumes  that  the  E.M.F.  is  maintained  absolutely  constant,  a  condition 
which,  as  will  be  seen  later,  can  be  realized  only  approximately.  In  the  general 
case  therefore  both  e  and  E  will  vary. 


FIELD-EXCITATION  OF  ROTARY  CONVERTERS        201 


power,  and  with  the  secondary  voltage  constant  at  the  brushes.  It  is 
the  same  problem  as  the  preceding  one  except  that  it  is  not  the  difference 
of  potential  at  the  terminals,  but  the  induced  E.M.F.  e,  which  must 
be  supposed  constant,  in  this  case. 

The  locus  of  the  point  C  is  again  the  right  line  CD  itself.  But, 
as  we  have  added  the  condition  that  the  segment  OB  remains  constant, 
the  point  O  will,  therefore,  remain  fixed  (Fig.  170). 


I" 


FIG.  170. 

For  each  position  of  the  point  C  the  corresponding  segment  CB 
indicates  the  current,  and  OC  indicates  the. E.M.F.  required  at  the 
generator.  Taking  the  E.M.F.  values  as  abscissas  and  the  current 
values  as  ordinates,  we  obtain  V-shaped  curves  (Fig.  18),  which,  in 


O    E.m.f.   of  Current-Supply  £ 

FIG.  18. 

consequence   of    making   £   constant,   are   independent  of  the   field- 
winding  and  of  the  law  of  saturation. 

It  is  more  interesting,  however,  to  take  the  excitation  ampere-turns 
as  abscissas,  as  in  the  case  of  synchronous  motors,  without  distinguish- 
ing those  due  to  the  series  and  the  shunt  windings.  This  amounts 
to  the  same  thing  as  assuming  the  machine  to  be  separately  excited. 


tJ-> 


202   GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

These  ampere-turns  are  determined  for  each  value  of  the  alternat- 
ing current  /  by  measuring  on  the  diagram  (Fig.  13),  the  correspond- 
ing segment  CD,  which  indicates  the  reactive  current  Id,  and  subtract- 
ing from  this  value  the  ampere-turns  corresponding  to  zero-load  which 
can  give  the  E.M.F.  e  and  the  magnetizing  ampere-turns  produced 
by  the  armature,  namely, 

A          (KN' 

Aa=( — = 

\  V  2 

using  the  +  or  the  —  sign  according  to  whether  Jd  is  positive  or 
negative,  i.e.,  according  to  whether  the  point  C  be  at  the  right  or  at 
the  left  of  D. 


FIG.  19. 

Taking  the  values  of  the  excitation  A ,  thus  determined,  as  abscissas, 
and  the  corresponding  values  of  /  as  ordinates,  we  obtain  a  V-curve 
such  as  shown  in  Fig.  19,  and  of  which  the  "  bend  "  corresponds  to 
the  load-condition  for  which  there  is  no  lag  of  the  current.  If  the  cor- 
responding values  of  E  are  also  plotted  as  ordinates,  a  second  V-curve 
will  be  obtained,  whose  bend  is  much  farther  to  the  right,  so  that, 
in  the  useful  portion  of  the  diagram,  it  consists  only  of  the  descending 
portion  of  the  curve.  These  curves  would  be  prolonged  to  the  left 
if  we  were  to  consider  negative  excitations. 

A  similar  pair  of  V-curves  will  be  obtained  for  each  value  of  the 
power-output  of  the  converter. 

Upper  Limit  of  Reactive  Current.  It  has  already  been  seen  that 
there  is  an  inferior  limit  to  the  reactive  current.  .  It  also  has  an  upper 
limit,  according  to  Eq.  (23),  namely,  that  value  which  makes  the  shunt- 


FIELD-EXCITATION  OF  ROTARY  CONVERTERS         203 

excitation  vanish  (a  negative  shunt-excitation  being  out  of  the  question). 
This  upper  limit  value 

:  I  .,   <,-*£ .  <,4> 

must  be  greater  than  the  minimum  limit,  unless  the  number  of  turns 
on  the  armature  is  excessive.  There  is  no  actual  need  of  an  upper 
limit  to  the  reactive  current,  because  the  aim  is  to  reduce  it  as  much 
as  possible,  on  account  of  the  undesirable  effects  produced  by  it  on 
the  distribution-system,  namely,  its  increasing  the  ohmic  losses,  and 
its  necessitating  a  higher  E.M.F.  E.  What  should  determine  the 
value  which  is  to  be  selected  for  i0  is  the  condition  that,  between  zero 
load  and  normal  load,  the  normal  variation  of  voltage  (which  deter- 
mines the  range  of  effect  produced  by  the  field-excitation  windings), 
should  be  that  which  is  suitable  for  the  practical  purpose  for  which 
the  machine  is  intended,  and  that  which  is  represented  by  the  current 
supply-characteristic  which  has  already  been  defined. 

The  investigation  of  this  point  is  facilitated  by  a  new  diagram, 
which  will  now  be  considered,  and  which  indicates  the  lag  characteristics 
of  the  converter  at  constant  potential. 

Lag-Characteristics  of  Rotary  Converters  at  Constant  Potential. 
Let  us  suppose  that,  in  some  way,  the  potential  difference  at  the  brushes 
of  the  converter  is  maintained  constant.  The  shunt-excitation,  which 
is  proportional  to  this  voltage,  will  also  remain  constant;  and  the  total 
ampere-turns  which  produce  this  voltage  will  also  remain  constant. 
We  will  therefore  have 

A  =  constant ;     Ad=  constant. 
Hence  A 8  +  Aa=  constant  (A—  Ad} (28) 

xkn  ,.  ,  KN' 

or  — —(Iw—  7o)H =Id=(A—  Ad) (29) 

V  2  V  2 

This  formula  simply  states  that,  with  constant  potential,  the  converter 
can  furnish  any  secondary  current  whatever  (proportional  to  Iw) 
which  passes  through  its  series-winding,  owing  to  the  automatic  decrease 
of  the  reactive  current,  Id,  as  the  latter,  of  itself,  takes  the  value  neces- 
sary to  make  up  the  same  total  magnetomotive  force  A . 

The  relation  between  IW—JQ  and  Id  is  simply  linear.  If,  therefore, 
in  Fig.  20,  the  load-values  be  represented,  as  previously,  by  distances 


204   GENERAL  DIAGRAMS   FOR  SYNCHRONOUS  MOTORS 

proportional  to  dD=-ZId  and  measured  along  the  reference-line 
and  if  the  corresponding  reactive  currents,  7<j,  be  represented  by  other 
segments  proportional  to  DC=ZId\  then  the  locus  of  the  point  C,  at 
the  end  of  the  segment  DC,  is  a  right  line,  QP,  whose  angular  coefficient 
with  respect  to  the  point  Q  as  a  pole  is 


DC     nkn 


(3o) 


This  coefficient  depends  only  on  the  field  and  armature  windings. 
The  right  line  QP  is  easily  drawn,  in  practice  by  determining  the  points 


Ac' 

C 


Q  and  P  which   correspond,  respectively,  to  zero-lag  and  zero-load: 
we  have 


^^^--~~~~~~ 

&--••'-   /*•- 
^/-%/           -,, 
/'"  ^"^-^ 

°*^                   ~€ 

FIG. 

=& 

20. 

For  the  point  P, 


r=.0=^_-^v- 


(31) 


For  the  point  Q, 


V  2 


(32) 


For  each  value  of  s  there  exists  a  value  of  A  —A^  This  value  is  obtained 
from  the  characteristic  curve  (Fig.  16),  by  drawing,  from  the  point 
m,  corresponding  to  this  coordinate,  the  right  line  ma,  making  the  angle 
d  with  respect  to  the  axis  of  A  values,  the  angular  coefficient  of  this 
line  being  the  resistance  of  an  average  turn  of  the  shunt  winding, 


tan  d=r. 


FIELD-EXCITATION  OF  ROTARY  CONVERTERS        205 

Therefore,  to  each  value  of  e  there  corresponds  a  right  line  PQ,  which 
is  more  or  less  distant  from  the  point  d\  but  all  these  right  lines  are 
parallel  to  each  other,  having  the  same  angular  coefficient,  p,  which 
depends  on  the  windings  only.  The  smaller  the  number  of  turns 
of  the  series-  winding,  the  farther,  evidently,  the  point  Q  will  be  from 
the  point  d,  on  the  right  line  BY.  When  there  is  no  series-  winding 
(in  the  case  of  a  shunt-wound  rotary  converter),  the  right  line  PQ 
becomes  a  right  line  parallel  to  QY,  drawn  through  P.  Conversely, 
the  decrease  of  the  number  of  turns  in  the  armature  displaces  the  point 
P  toward  the  right,  on  the  perpendicular  dP. 

Finally,  increasing  or  decreasing  the  shunt  ampere-turns  decreases 
or  increases  the  difference  A—  Ad,  and,  consequently,  brings  the 
right  line  PQ  nearer  to  or  farther  from  the  point  d.  This  line  passes 
through  the  point  d  when  Ad=A. 

Effective  Characteristic  of  Rotary  Converter  under  Load. 
Following  the  methods  just  described,  it  is  easy  to  study,  on  the  diagram, 


{-" 

t  ,' 


_ 

~o^o"  ~o'"o  &  B:  P--\^ 

I  p  r 

FIG.  21, 

the  variations  of  voltage  at  the  terminals  of  the  converter,  as  a  function 
of  its  current-output.  It  is  only  necessary  to  reverse  the  problem 
and  to  find  the  output  corresponding  to  each  voltage,  writhin  the  limits 
of  variation  prescribed  or  allowed  for  the  voltage. 

Let  e,  ef,  e",  .  .  .  represent  a  series  of  values  of  the  voltage  e.  Let 
these  values  be  represented,  in  Fig.  21,  by  the  corresponding  distances 
BO,  BO',  BO".  .  .  .  From  the  points  O,  O',  O"  ',  ...  let  circular  arcs 
be  described,  with  a  radius  equal  to  E,  if  the  supply  E.M.F.  is  constant, 
or  equal  to  E,  E',  E".  .  .  respectively,  if  the  supply  E.M.F.  is  variable. 
Let  the  points  of  intersection  C,  C',  C",  ...  of  these  circles  with  the 
lag-characteristic  line  QP,  Q'P',  Q"P".  .'  .  correspond,  respectively, 
to  the  E.M.F.  values  e,  ef,  e",  ....  These  points  will  define  the  cor- 


206   GENERAL  DIAGRAMS   FOR  SYNCHRONOUS  MOTORS 

responding  current  values,  Iw  and  Id,  of  the  converter,  and,  in  particular, 
the  loads  dD,  dD',  dD".  ...  If  the  required  characteristic  has  already 
been  drawn  by  points,  for  the  same  values  of  e,  e',  e",  .  .  .,  the  compar- 
ison of  the  two  curves  will  show  at  once  the  discrepancies  between 
the  values  wanted  and  those  which  the  apparatus  can  give. 

The  objection  to  this  diagram  is  that  the  circles,  in  practice,  cut  the 
QP  right  lines  at  very  sharp  angles,  and  that  the  points  of  intersection 
are  somewhat  uncertain,  i.e.,  it  is  difficult  to  locate  them  accurately. 

In  order  to  make  the  comparison  with  greater  precision  by  calcula- 
tion, the  value  of  Iw  is  calculated,  as  a  function  of  e,  by  Eqs.  (2).  and 
(29),  namely, 

E2-e*,    .     .     .       (2) 


These  two  equations,  containing  two  unknown  quantities,  /„,  and  Id, 
enable  Id  to  be  eliminated,  and  a  solution  to  be  obtained  for  /«,  in  terms 
of  known  quantities,  £  being  supposed  to  have  been  previously  deter- 
mined. This  calculation,  though  a  trifle  tedious,  presents  no  difficulties. 

The  conditions  of  operation  in  various  cases  can  be  easily  pre- 
determined without  drawing  the  curve  accurately.  Since,  from  what 
precedes,  the  angle  of  the  PQ  phase-lines  depends  on  the  reactive 
current  iy  permitted  at  zero  load,  it  will  be  apparent  that  the  choice 
of  values  for  this  current  should  be  made  with  the  idea  of  making  the 
actual  curve  agree  as  closely  as  possible  with  the  desired  curve.  The 
principal  practical  cases  will  be  investigated  from  this  standpoint. 

Application  in  the  Case  of  Separate  Excitation.  Let  us  suppose, 
at  first,  for  the  sake  of  greater  simplicity,  that  the  shunt-excitation 
is  replaced  by  a  separate  excitation  which  is  constant.  Ad  is  then 
constant.  It  is  seen,  immediately,  that  if  no  series- wind  ing  is  added, 
the  converter  can  only  give  a  decreasing  voltage  as  the  load  increases. 
The  lag-characteristic  for  all  loads  is,  in  fact,  a  right  line,  PQ,  parallel 
to  BY  (Fig.  22);  and  if  we  take,  on  that  line,  points  C'C",  whose 
distance  from  P  increases,  the  circles  of  constant  radius,  E,  cut  the 
axis  BO  at  points  O,  O',  O",  which  come  nearer  to  B;  hence,  the  volt- 
ages proportional  to  OB  are  decreasing. 

Therefore,  whatever  may  be  the  reactance  and  the  value  selected 
for  the  external  E.M.F.  E,  it  is  not  possible  to  maintain  the  voltage 
constant  at  the  brushes  on  the  direct-current  side,  with  a  separate  excita- 
tion, and  still  less  so  with  a  shunt-excitation.  It  goes  without  saying 


FIELD-EXCITATION  OF  ROTARY  CONVERTERS        207 

that  the  best  excitation  under  these  conditions  of  operation  with  falling 
voltage  would  be  one  which  neutralizes  the  reactive  current  dP  at 
all  loads;  but  this  mode  of  operation  is  not  of  great  practical  interest. 
It  has  been  proposed  to  use  converters  with  a  falling  voltage-char- 
acteristic in  connection  with  storage  batteries,  so  as  to  make  the  batteries 
discharge  when  the  load  is  heavy,  and  make  them  charge  when  the  load 
is  light.  With  the  usual  method  of  current-supply  this  mode  of  opera- 
tion would  cause  variations  of  TO,  20  per  cent,  or  more,  in  the  supply 
voltage.  Dr.  F.  B.  Crocker 1  has  devised  an  ingenious  method  of  regulat- 
ing the  supply-voltage  which  enables  a  wide  range  of  E.M.F.  regulation 
to  be  obtained  in  a  rotary  converter,  either  with  a  falling  or  a  rising 

IY 


-  /     -----'/c' 

//. 

18  ""--.     ' 


FIG.  22. 

characteristic,  without  affecting  the  E.M.F.  of  the  source  of  supply. 
The  same  result  is  also  accomplished  by  the  so-called  "  split-pole  " 
rotary  converter,  in  another  way. 

If  we  add  a  series-winding  which  causes  the  right  line  PQ  to  incline 
toward  the  left,  the  variations  of  e  will  evidently  become  smaller.  We 
proceed  to  find  the  condition  which  reduces  these  variations  of  £  to  a 
minimum,  i.e.,  which  makes  the  converter  give  a  substantially  constant 
voltage. 

It  is  evidently  necessary  that  the  lag-characteristic  PQ  should 
coincide  as  far  as  possible,  within  the  practical  load  limits,  with  a  circular 
arc  described  from  the  point  O  as  a  center  situated  at  the  end  of  the 
line  BO=E.  Let  F  (23)  represent  the  zero-load  condition;  let  G 
represent  the  average  load,  and  M  the  maximum  load.  A  right  line 

1  See  U.  S.  Patents  to  F.  B.  Crocker,  on  "Automatic  Regulation  of  Rectifiers 
and  Rotary  Converters,"  No.  891,797,  June  23,  1908,  and  No.  1,012,524, 
Dec.  19,  1911. 


208   GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

sab  can  be  drawn  which  cuts  the  circle  at  two  intermediate  load-points 
ab  between  the  extreme  points  F  and  M,  in  such  a  way  that  this  line 
will  not  be  far  distant  from  the  circular  arc  of  any  point  in  its  useful 
portion,  PM.  The  segments  dP  and  dQ,  read  off  from  the  diagram 
(by  the  proper  scale)  will  determine  the  currents  7*0  and  /„,  correspond- 
ing to  zero-load  and  normal  load;  and  Eqs.  (31)  and  (32)  can  be  used 
with  these  values  to  deduce  the  number  of  constant  ampere-turns, 
Ad,  which  are  necessary,  and  also  the  number  of  turns  n  of  the  series- 
windings. 

The  slight  residual  differences  in  the  E.M.F.  £  will  be  obtained  by 
drawing  circles  of  radius  E  from  every  point  on  the  right  line  PS  and 


taking  their  intersections  with  the  axis  of  voltages  BO.  The  value  of 
the  reactance  X  affects  the  precision  of  regulation,  because,  the  greater 
the  length  of  the  arc  FM,  corresponding  to  a  given  variation  of  load, 
the  more  difficult  it  is  to  make  it  coincide  with  a  right  line  such  as  PS. 
To  improve  the  regulation,  therefore,  the  segment  dD=ZIw,  which 
measures  the  maximum  load  Iw,  ought  to  be  decreased  to  a  minimum; 
and,  to  this  end,  the  reactance  (which  is  the  more  important  factor  in 
the  value  of  Z)  ought  to  be  decreased  to  the  lowest  value  possible. 
However,  as  already  seen,  the  reactance  cannot  be  decreased  below 
a  certain  value  A'0  without  causing  the  zero-load  reactive  current  i0  to 
increase.  It  is  therefore  necessary  to  take  a  suitable  mean  value  be- 
tween Xo  and  zero,  which  will  depend  on  the  circumstances  and  the 
conditions  of  the  case.  In  general,  the  more  the  load  of  the  converter 
is  uniform  and  the  less  often  it  runs  without  load,  the  more  it  will  be 
possible  to  reduce  X. 


FIELD-EXCITATION  OF  ROTARY  CONVERTERS        209 

To  overcompound  the  converter  the  right  line  PQ  should  be 
straightened  more,  by  raising  the  point  O  higher  on  the  right  line  BY; 
but  the  principle  always  remains  the  same,  namely,  to  draw  the  right 
line  PQ  in  such  a  way  that  it  shall  come  as  close  as  possible,  at  every 
point,  to  the  desired  characteristic,  which  should  itself  be  drawn 
first. 

Application  to  the  General  Case  of  Self-Excitation.  In  the 
general  case  of  self-excited  converters,  the  solution  is  nearly  the  same, 
but  it  is  complicated  by  the  fact  that  the  shunt-excitation  varies  with  the 
direct-current  voltage.  Hence,  instead  of  only  one  right  line  PQ  to 
indicate  the  lags,  there  will  be  a  series  of  parallel  right  lines  correspond- 
ing to  various  values  of  e. 


o1       o 


FIG.  24. 

In  the  case  of  constant  potential  regulation,  the  method  of  deter- 
mining the  excitation-windings  and  the  reactance  of  the  circuit  remains 
exactly  the  same  as  given  in  the  preceding  paragraph,  since  the  shunt- 
winding  must,  by  hypothesis,  produce  constant  excitation.  But  the 
small  variations  of  Ad  have  to  be  considered  in  the  final  calcula- 
tion of  the  differences  of  voltage  due  to  the  imperfection  of  the 
regulation. 

For  values  of  £  lower  than  the  normal  value,  the  shunt-excitation 
decreases  of  itself  and,  consequently,  the  right  line  PQ  comes  closer 
to  d,  remaining  parallel  to  itself.  On  the  other  hand,  this  line  recedes 
from  d  when  e  increases.  It  is  easy  to  see,  in  Fig.  24,  that  this  effect 
tends  to  increase  the  voltage-differences  as  compared  with  these  dif- 
ferences when  the  shunt-excitation  is  constant. 

When  the  machine  is  to  be  overcompounded  the  zero-load  and 
the  normal  load  no-lag  conditions  determine,  as  before,  the  reactive 


210    GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

ampere-turns,  for  zero-load,  AQ,  and,  therefore,  the  shunt  and  series 
windings 

Let  e  and  e'  represent  the  voltage-values  corresponding  to  these 
two  load-conditions,  according  to  the  excitation-curve.  The  lag- 
lines  corresponding  to  all  the  intermediate  values  are  parallel,  and  are 
comprised  between  two  lines  which  are  determined  by  Eq.  (29)  when  the 
right-hand  member  is  made  equal  to  AI—  A0  and  to  A',  respectively. 
These  lines  pass  through  P  and  P'  respectively. 

In  order  to  obtain  effective  overcompounding,  i.e.,  to  have  e'  >  e,  it 
is  necessary  that  the  point  O  should  not  be  very  far  distant  from  P 
on  the  right  line  of  reference  BY.  This  means,  simply,  that  it  is  nec- 
essary to  have  either  a  powerful  series-winding,  or  else  a  large  reactive 
current  with  zero-load.  The  first  of  these  two  ways  appears  to  be  the 


5-<f       — 

X  --------  ;  ---  g' 


Fig.  25. 

preferable  one;  but  inasmuch  as  it  is  always  desirable  to  increase  the 
inclination  of  the  right  line  B  Y  to  the  right  in  order  to  decrease  e,  it  is 
necessary  to  resort  to  both  plans  to  obtain  overcompounding. 

Regulation  of  Supply  E.M.F.  by  Compounding  of  the  Generator. 
As  already  stated,  it  is  possible,  theoretically,  to  do  away  with  the  reac- 
tive current  at  all  loads  by  using,  as  the  source  of  current-supply  for  the 
converter,  a  suitably  compounded  alternator,  giving  an  E.M.F.  E, 
which  increases  with  the  load  in  such  a  way  that  the  point  C  is  always 
displaced  on  the  right  line  of  reference  BY,  while  e  remains  constant 
or  else  increases,  according  to  the  results  desired.  It  is  interesting  to 
know  how  the  excitation  should  be  effected  in  such  a  case. 

If  it  be  required  to  have  the  direct-current  voltage,  e,  constant,  the 
excitation  will  evidently  be  that  of  a  simple  shunt-winding;  and  its 
value  is  easy  to  determine  by  the  condition  that  it  must  give  exactly  the 
voltage  e,  with  zero-load,  since  there  is  not  to  be  any  reactive  current. 


FIELD-EXCITATION  OF  ROTARY  CONVERTERS        211 

If,  on  the  other  hand,  it  be  required  to  have  the  voltage  rise  with 
the  load,  the  excitation  must  be  compound.  The  shunt- winding  will 
still  be  such  that  it,  alone,  gives  the  no-load  E.M.F.  e,  and  the  series- 
winding  will  be  such  that  it  will  produce  the  additional  ampere-turns 
necessary  to  give  the  E.M.F.  e'  required  with  the  load  Iw.  As  already 
stated,  in  a  .case  like  this,  it  is  desirable  to  decrease  the  reactance  X, 
and  the  lag  0,  to  a  minimum. 

Regulation  of  Voltage  by  Varying  the  Reactance  X  in  the 
Circuit.  The  two  windings,  in  this  case,  will  be  determined  in  the  same 
way  as  in  the  preceding  case,  since  the  voltage  of  the  alternating  cur- 
rent supplied  to  the  converter  is  still  regulated  by  outside  means  without 
causing  lag;  in  the  current  inside  the  machine.  The  excitation  will 
be  that  produced  by  a  simple  shunt-winding  when  constant  voltage 
is  wanted,  and  that  produced  by  a  compound-winding,  when  a  rising 
voltage  is  wanted.  The  shunt-winding  will  be  determined  by  the 
voltage  required  with  no  load,  and  the  series-winding  will  be  determined 
by  the  voltage  required  with  full  load. 

Possibility  of  Suppressing  the  Shunt-Winding.  It  is  not  abso- 
lutely necessary  to  use  either  a  separate  exciting  winding,  or  else  a 
shunt  winding,  as  has  been  supposed  hitherto;  because,  by  making 
the  number  of  armature-winding  turns  greater  and  decreasing  the  air- 
gap,  it  is  always  possible  to  obtain  the  necessary  no-load  excitation, 
merely  by  the  effects  of  the  armature-reaction.  Eqs.  (21)  and  (22),  in 
that  case,  reduce  to  the  following  forms: 

nkn  ,, 

—7=(Im-Jo)  =  Am (2ia) 

A/2 

KN'  . 

—- =*o=40 (220) 

V  2 

It  is  not  possible  to  obtain  constant  voltage  or  rising  voltage,  in 
the  direct-current  circuit  with  this  method  any  more  than  when  a  shunt- 
winding  is  used,  without  series-winding.  This  method  has,  to  some 
extent,  the  self-regulating  properties  of  transformers,  but  to  a  degree 
which  is  inadequate,  owing  to  the  greater  reluctance  of  the  magnetic 
circuit. 

On  the  other  hand,  the  high  values  which  the  reactive  currents 
must  then  be  allowed  to  attain  have  the  great  objection  of  causing 
increased  heating  of  the  armature  by  increasing  the  total  current  in 


212   GENERAL  DIAGRAMS   FOR  SYNCHRONOUS  MOTORS 

the  armature;  and  also  of  increasing  the  lag  of  the  current  in  the 
supply-circuit,  thereby  reducing  the  output-capacity  of  the  generator, 
increasing  the  voltage  drop  in  the  line,  etc.  In  a  word,  this  method 
has  all  the  faults  and  objections  due  to  heavy  reactive  currents,  in  addi- 
tion to  which  the  commutation  of  the  machine  is  much  less  satisfactory, 
there  being  more  tendency  to  sparking,  owing  to  the  shortened  air-gap. 

This  method,  which  was  used  by  certain  concerns,  has  now  been 
abandoned  altogether.  It  is  referred  to  here  merely  to  show  the  ease 
with  which  it  is  possible  to  calculate  the  excitation  for  the  conditions, 
obtaining  in  this  case  as  well  as  in  the  preceding  cases.  This  method 
has  one  advantage,  however,  that  of  making  the  satisfactory  performance 
of  the  converter  more  independent  of  the  form  of  the  E.M.F.  curve  of 
the  generator. 

Conclusion.  It  has  been  shown,  in  what  precedes,  "that  the  vector- 
diagram  for  synchronous  motors,  when  referred  to  two  axes,  lends  itself, 
as  the  result  of  natural  simple  extensions,  to  the  study  of  the  rotary 
converter  and  enables  its  conditions  of  operation  to  be  analyzed  for 
the  most  varied  cases.  This  method  of  analysis  serves  to  render  more 
intelligible  the  very  complex  problem  involved  in  the  voltage  regula- 
tion of  the  rotary  converter,  which  is,  perhaps,  the  most  complicated 
problem  presented  in  connection  with  alternating-current  machinery. 

The  methods  set  forth  in  the  preceding  pages  not  only  enable  the 
phenomena  to  be  foreseen  qualitatively,  but  it  also  enables  them  to  be 
calculated  numerically,  by  combining  the  diagrams,  or  the  equations 
which  are  derived  from  them,  with  the  data  obtained  from  experiments 
or  tests. 


CHAPTER  III 
STABILITY  OF  OPERATION   OF  ROTARY  CONVERTERS 

WHEN  in  operation,  rotary  converters  sometimes  produce  variations 
of  voltage  above  and  below  their  normal  voltage;  and  they  may  also, 
at  times  spark,  or  even  flash,  at  the  commutator.  These  phenomena 
might  be  attributed  to  changes  of  frequency  due  either  to  variations  of 
angular  velocity  of  the  prime  mover  driving  the  alternator  which 
supplies  current  to  the  converters,  or  else  due  to  oscillations  of  its  speed- 
governor. 

To  eliminate  these  two  causes  of  irregular  operation,  the  follow- 
ing experiment  was  tried:  the  rotary  converter  was  driven  as  a  motor 
by  current  taken  from  a  storage  battery,  which  also  excited  the  shunt- 
field  winding.  Three-phase  currents  taken  from  the  A.C.  side  of  the 
machine  were  passed  through  a  raising  transformer  to  raise  their  voltage. 
The  voltage  was  then  lowered  by  step-down  transformers  and  the 
energy  transmitted  through  them  was  absorbed  by  means  of  three 
identical  rheostats.  The  arrangement  was  equivalent,  practically, 
to  connecting  a  three-phase  inductive  resistance  to  the  secondary  of 
the  first  transformer  (Fig.  26).  It  was  then  observed  that,  on  increasing 
gradually  the  load  on  the  converter,  a  critical  point  was  reached  where 
its  speed  began  to  oscillate  between  limits  which  were  all  the  farther 
apart  the  more  the  load  was  increased. 

To  explain  this  phenomenon,  let  us  note  that  the  converter  was 
sending  reactive  currents  into  the  circuit,  and  that  these  reactive  cur- 
rents tended  to  weaken  the  magnetic  field.  Now  in  the  case  of  a  load 
which  is  below  a  certain  value  (which  we  will  term  the  limiting  or  critical 
load)  the  strength  of  this  reactive  current  should  decrease  when  the 
speed  of  the  rotary  converter  increases,  and  reciprocally,  it  should 
increase  when  the  speed  decreases.  The  contrary  effect  should  be 
produced  when  the  load  exceeds  this,  critical  load. 

To  demonstrate  this,  let  us  refer  to  a  transformer  diagram  of  the 
kind  which  takes  magnetic  leakage  into  consideration.  It  is  known 
(Fig.  27)  that  if  OF  is  the  axis  of  E.M.F.'s  the  end  of  the  vector  (OM) 

213 


214  GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 


representing   the   strength   of  the  primary  current  describes  a  semi- 
circle which  cuts  the  axis  of  x  at  two  points,  A  and  B,  such  that  OA=--i<iv 


represents  the   zero-load  magnetizing    current,   and   OB=—,   where 

a 

a  =  --  ,  in  which  v\  and  v%  are  the  leak  age  coefficients  for  the  two  wind- 


ings  of  the  transformer. 

By  hypothesis,  the  voltage  of  the  source  of  current-supply  is  con- 
stant. Therefore,  an  increase  of  speed  of  the  converter,  by  causing  an 
increase  in  the  frequency,  will  bring  about  a  decrease  of  the  magnetiz  - 
ing  current  corresponding  to  zero-load.  We  will  have  a  new  circle, 


Rotary  Converter 


Three-Phase  Transformer 


Three-Phase 

Inductive 

Resistance 


FIG.  26. 


A 'Bf,  whose  diameter  will  be  i'dv\  —  —  i  )•     The  two  circles  will  inter- 


sect each  other  at  a  point  M,  which  is  on  the  line  drawn,  from  O, 
tangent  to  one  of  the  circles. 

It  is  seen,  at  once,  on  the  diagram,  that  the  right  line  ON,  which 
represents  the  active  current  in  relation  to  the  primary  current  OM, 
indicates  the  critical  load  already  mentioned.  In  fact,  for  the  value 
OH,  corresponding  to  a  load  which  is  lighter  than  the  critical  load,  i.e., 
which  makes  OK  <  ON,  it  is  seen  that  the  reactive  current  has  decreased 
from  the  value  OL  to  the  value  OL\,  as  the  result  of  a  speed-increment. 
On  the  contrary,  for  the  current  OP,  corresponding  to  a  load  which 
is  heavier  than  the  critical  load,  i.e.,  which  makes  OQ  >  ON,  the  reactive 
current  has  increased  from  OR  to  OR\.  It  is  now  easy  to  analyze  the 
phenomena  observed. 

In  the  first  place,  when  the  load  was  lighter  than  the  critical  load 
of  the  machine,  an  increment  of  speed  produced  a  decrease  of  the 
reactive  current,  and,  consequently  a  decrease  in  demagnetizing  action; 
and,  since  the  converter  was  running  as  a  direct  current  motor,  with 
constant  shunt-field-excitation,  its  total  excitation-flux  was  increased. 
Therefore,  the  E.M.F.  on  the  D.C.  side  being  constant,  the  speed  of 


STABILITY  OF  OPERATION  OF  ROTARY  CONVERTERS     215 

the  converter  had  to  decrease,  in  order  to  maintain  the  same  induced 
E.M.F.  The  converter  therefore  varied  in  speed  in  order  to  maintain 
the  induced  E.M.F.  constant. 

In  the  second  place,  when  the  load  was  heavier  than  the  critical 
load  of  the  machine,  an  increase  of  reactive  current  was  produced 
by  a  speed  increment.  Hence,  there  was  a  decrease  in  the  total  excita- 
tion flux.  The  speed  of  the  motor  therefore  tended  to  increase  until 
the  current  supplied  from  the  battery  was  limited  by  the  resistance  of 
the  circuits.  As  the  armature  of  the  converter  was  then  in  a  state  of 

0         A' ALL        I          R     R.     Axis  of  Reactive  Currents        B B         X 


FIG.  27. 


unstable  dynamic  equilibrium  its  speed  soon  began  to  decrease;  and 
thus  speed-oscillations  were  produced. 

In  practice,  therefore,  it  is  desirable  that  the  load  put  on  the  rotary 
converter  should  not  exceed  the  critical  load  above  mentioned. 

Let  us  see,  now,  what  can  be  done  to  raise  this  load-limit.  We 
will  use  the  same  diagram  as  before  (Fig.  27). 

We  see  readily  that  what  is  necessary  is  to  increase  the  vector 
ON.  This  may  be  accomplished  in  different  ways: 

First,  if  the  frequency  is  decreased,  then,  for  a  given  voltage,  other 
things  being  equal,  the  magnetizing  current  will  have  to  be  increased. 

Therefore  the  diameter  of  the  circle  corresponding  to  i^v  (  —  —  i  )  will 
be  increased,  and,  consequently,  the  point  of  contact  M,  of  the  line 


216   GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

which  is  tangent  to  this  circle,  is  farther  away  from  the  axis  Ox;  hence 
ON  is  increased. 

Second,  if  the  coefficient  of  self-induction  of  the  transformers  is 
decreased,  while  the  other  quantities  remain  unchanged,  the  magnetiz- 
ing current  corresponding  to  zero-load  will  increase;  hence  ON  is 
increased. 

Third,  if  the  magnetic  leakage  is  decreased,  then  —  increases,  and, 

a 

consequently,  the  diameter  of  the  circle  also  increases,  causing  as  before 
the  point  M  to  move  farther  away  from  Ox,  and  increasing  ON. 

To  sum  up,  then,  the  limiting  load  on  a  rotary  converter  can  be 
increased:  i,  by  diminishing  the  frequency  of  the  A.C.  source;  2, 
by  reducing  the  self-induction  of  the  transformers;  3,  by  reducing  the 
magnetic  leakage. 

These  facts  have  been  all  verified  experimentally. 

II.  Let  us  now  consider  the  rotary  converter  as  a  transformer  of 
alternating  into  direct  current. 

The  rotary  converter  used  in  the  previous  experiments  was  operated 
regularly  at  full  load  with  the  transformer  connected  to  a  suitable 
source  of  alternating  current  supply. 

In  order  to  regulate  the  voltage  of  the  direct  current  produced  by 
the  converter,  an  (A.C)  induction-regulator  having  a  high  coefficient 
of  magnetic  leakage  was  included  in  the  primary  circuit  of  the  trans- 
former. The  rotary  converter  then  became  subject  to  speed  oscilla- 
tions of  long  period  as  soon  as  its  load  was  increased  to  half  the  load 
which  was  carried  without  the  induction- regulator.  To  explain  this 
fact  we  shall  show  that  every  speed-increment  of  the  rotary  converter 
causes  an  increase  in  the  difference  of  potential  at  its  terminals. 

Let  us  refer  again  to  the  converter-diagram.  Let  OA  (Fig.  28), 
represent  the  difference  of  A.C.  potential  at  the  terminals  of  the  con- 
verter; OC,  the  supply-voltage  which,  in  this  case,  is  constant;  0, 
the  phase-angle  between  these  two  E.M.F.'s;  AD,  the  active  current, 
including  the  current  corresponding  to  friction,  eddy  currents,  etc., 
DC,  the  reactive  current,  counted  as  positive  on  the  right  side  of 
the  axis  of  currents  BY,  and  counted  as  negative  on  the  left  side 
thereof. 

Every  increment  of  speed  of  the  converter  causes  a  decrease  of  the 
angle  0.  Consequently  the'  diagram  shows  that,  OC  being  con- 
stant, the  active  current  AD  will  decrease  (to  AD')  and  the  reac- 
tive current  (DC)  will  increase  (to  D'C')-  Moreover,  as  in  the 


STABILITY  OF  OPERATION  OF  ROTARY  CONVERTERS    217 

case  of  synchronous  motors,  there  appears  an  E.M.F.  which  is  -  in 

2 

advance  of  the  E.M.F.  e  and  which  is  equal  to  coL'4(Iw).     But  AIW 
is  itself  negative    (being  equal  to  AD'— AD=  —  DD'}\  therefore,  this 

E.M.F.  (AF)  is  -  behind  the  E.M.F.  e.     This  E.M.F.  here  tends  to 

2 

diminish  the  increment  of  /<*,  but  it  increases  Iw  still  more. 

The  ampere-turns  by  which  the  E.M.F.  e  is  induced  are  equal  to 

KN' 

Adi  — =  Id,   in  which   Ad  represents  the   ampere-turns  due    to  the 

V2 

constant  excitation.     It  is  seen  at  once,  that  whether   the   point  C 
be  at  the  right  or  at  the  left  of  the  line  of  reference  (BY)  for  current- 


Axis  oF  Reference  for  EMF's. 
r 


FIG.  28. 

values,  the  value  of  the  excitation  for  an  increment  of  Id  is  always 

KN' 

equal  to  Aa-\ — :— J(/d);    consequently,  the  excitation  will  increase  in 

V  2 

all  cases,  and  so  will  the  E.M.F.  s. 

The  power  developed  at  the  shaft  of  the  rotary  converter,  when 
in  normal  operation  is  equal  to 


When  the  increment  of  speed  occurs,  the  increase  of  power  developed 
at  the  shaft  is  equal  to 


In  order  to  have  stability  of  operation  we  should  have 


218    GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

It  is  seen,  from  the  diagram,  that,  at  light  loads,  0  being  then  small, 
the  increment  of  Id  is  small  in  comparison  with  the  increment  of  Iw, 
whereas  with  heavy  loads,  0  being  then  large,  the  reverse  is  the  case. 

The  observed  phenomenon  can  now  be  explained. 

If  the  voltage  at  the  terminals  of  the  converter  remained  constant 
the  torque  would  decrease  when  the  speed  increases.  But  each  increase 
of  voltage  causes  an  increase  of  torque.  At  light  loads  the  first  effect 
is  greater  than  the  second;  therefore,  since  the  torque  decreases, 
there  is  an  automatic  regulation  of  the  speed. 

Beyond  a  certain  load,  the  second  effect  preponderates,  and  there 
is  an  increase  in  the  torque.  The  converter  therefore  continues  to 
gain  in  speed  until  the  torque  ceases  increasing.  It  is  then  in  a  state 
of  unstable  equilibrium  and  it  then  loses  speed  until  the  torque  ceases 
decreasing. 

Experiment  shows  that  the  amplitude  of  the  speed-oscillations 
increases  rapidly  with  the  load  and  that  it  then  becomes  impossible 
for  the  machine  to  operate.  The  oscillations  can  be  diminished  by 
taking  advantage  of  the  fact  that  the  torque  of  the  converter  diminishes 
with  its  excitation.  If  we  take  a  compound  excitation  comprising  a 
shunt-winding  connected  to  the  terminals  of  the  converter  and  a  series- 
winding  connected  differentially,  it  will  be  seen  that  whenever  there 
is  an  increase  of  voltage  at  the  terminals  of  the  converter,  there  will 
be  an  increase  of  the  load-current.  This  current,  acting  upon  the 
series-winding,  will  cause  an  increase  in  the  series-excitation  which  is 
opposed  to  the  shunt-excitation.  Therefore  the  total  excitation  will 
decrease,  and  the  torque  will  also  decrease.  These  different  effects 
can  be  calculated  and  adjusted  so  as  to  balance  or  overbalance  the 
increase  of  torque.  Actual  test  shows  that  this  can  be  done,  and  that 
the  converter  can  be  made  to  carry  full  load  even  when  the  induction- 
regulator  is  used,  if  the  series-winding  is  reversed.  It  appears,  there- 
fore, that,  in  the  case  of  a  rotary  converter  which  is  transforming 
direct  current  into  alternating  current,  it  is  desirable  to  have  a  com- 
pound winding  in  which  the  shunt  and  series  excitations  add  themselves, 
or  act  cumulatively. 

On  the  other  hand,  a  rotary  converter  serving  to  transform  alternat- 
ing current  into  direct  current  should  have  a  compound  winding  in 
which  the  series  and  shunt  excitations  act  differentially. 


CHAPTER    IV 
OPERATION   OF  SEVERAL  ROTARY    CONVERTERS  IN    PARALLEL 

THE  parallel  working  of  rotary  converters  is  a  much  more  difficult 
matter  than  the  parallel  working  of  ordinary  generators,  owing  to  the 
fact  that  the  load  has  a  tendency  to  distribute  itself  unequally,  without 
any  correcting  action  in  the  case  of  rotary  converters. 

Any  increase  of  speed  occurring  in  a  converter  causes  the  alternating 
current  to  lead  in  phase  with  respect  to  the  induced  E.M.F.  This 
strengthens  the  magnetic  field  of  the  converter,  and,  consequently, 
causes  the  induced  E.M.F.  to  be  still  further  increased.  Therefore, 
there  is  a  tendency  to  unstable  operation.  To  avoid  the  unequal 
distributions  of  load  in  rotary  converters  which  are  connected  in  parallel, 
they  not  only  require  careful  attendance,  but  their  design  and  con- 
struction demand  much  care,  because  the  various  converters  intended 
for  parallel  working  should,  as  nearly  as  possible,  have  the  same 
armature-resistance  and  the  same  magnetic  reluctance  in  the  magnetic 
circuits  of  the  field-magnets;  in  fact,  the  magnetic  leakage  and  the 
magnetization  curve  should  be  substantially  the  same  for  all  the 
machines. 

It  is  often  necessary  to  give  up  the  attempt  to  compound  the  rotary 
converters  and  to  use  only  shunt-wound  rotary  converters,  for  parallel 
working.  In  any  case  when  compound  windings  are  used  it  is  almost 
always  necessary  to  adjust  these  windings  after  the  machines  are  in 
position,  in  order  to  make  them  run  in  balance  at  all  loads.  It  is 
understood,  of  course,  that  an  "  equalizer  "  connection  is  necessary 
between  the  different  machines  as  in  the  case  of  all  other  direct-current 
compound-wound  dynamos  connected  in  parallel. 

It  is  more  difficult  still  to  make  the  converters  run  in  balance  when, 
instead  of  being  coupled  directly,  they  are  coupled  through  feeders 
supplying  current  to  a  line  which  is  common  to  all  of  them,  as,  for 
instance,  a  section  of  trolley  or  of  third-rail  conductor,  in  electric 
railway  operation.  In  such  cases  it  has  been  found  necessary  to 

219 


220    GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

• 

reduce  the  voltage-drop  in  the  feeders  to  a  small  proportion,  i.e.,  2  to 
3  per  cent  of  the  normal  working  voltage. 

INHERENT    OSCILLATIONS    OR    PUMPING    OF    CONVERTERS 
CONNECTED   IN  PARALLEL 

Speed-oscillations  can  be  produced  in  rotary  converters,  same 
as  in  synchronous  motors,  by  periodic  speed-variations  in  the 
alternators  supplying  them  with  current,  when  these  alternators  are 
driven  by  steam  engines  whose  angular  velocity  varies  at  different 
portions  of  each  piston-stroke ;  and  they  may  also  be  produced  by  sud- 
den changes  or  fluctuations  of  load.  These  speed-oscillations  in 
rotary  converters  may  even  have  a  much  greater  amplitude  than  in 
other  machines,  because  they  are  not  opposed  by  a  contrary  variation 
of  the  torque  acting  upon  the  armature-shaft.  As  already  stated, 
when  the  speed  of  a  converter  increases,  its  current  is  thrown  out  of 
phase  and  becomes  leading  with  respect  to  the  induced  E.M.F.  This 
causes  the  magnetic  field  to  become  stronger  and  the  induced  E.M.F., 
to  rise;  consequently,  the  secondary  (direct  current)  output  increases. 
The  primary  (input)  power  increases,  however,  in  substantially  the  same 
proportion;  hence,  there  is  no  other  variation  in  the  torque  acting 
upon  the  armature  shaft  than  that  which  can  result  fr*m  the  varia- 
tions of  the  losses  with  the  load;  and  this  effect  is  always  quite  small 
in  comparison  with  the  inertia  of  the  armature. 

The  amplitude  of  the  oscillations  may,  therefore,  be  very  large; 
sometimes  it  is  sufficient  to  make  the  converter  "  fall  out  of  step." 
More  often,  its  effects  are  made  evident  by  fluctuations  in  the  (D.C.) 
E.M.F.  of  the  converter — which  have  amounted  to  as  much  as  30  per 
cent — and  also  by  sparking  at  the  commutator. 

The  E.M.F.  fluctations  are  easily  explained  by  the  variations  of 
speed  and  reactive  current  which  occur  simultaneously  with  them. 
The  sparking  is  due  to  a  much  more  complicated  action.  Owing  to 
the  inertia  of  the  armature,  every  increase  in  speed,  causing  additional 
kinetic  energy  to  be  stored  in  the  armature-mass,  makes  the  primary 
(input)  power  greater  than  the  secondary  (output)  power;  and  every 
decrease  in  speed,  causing  the  armature  to  lose  some  of  the  kinetic 
energy  stored  in  its  mass,  produces  the  contrary  effect  of  making  the 
secondary  power  greater  than  the  primary  power. 

In  both  cases  the  transverse  armature-reactions  produced  by  the 
primary  and  secondary  active  currents  no  longer  balance  each  other, 
and  the  axis  of  the  resultant  magnetic  field  oscillates  from  one  side  of 


SEVERAL  ROTARY  CONVERTERS  IN  PARALLEL   221 

the  line  (diameter)  of  commutation  to  the  other;  and,  as  the  brushes 
cannot  follow  these  oscillations,  sparking  occurs,  just  as  it  does  in  the 
case  of  a  dynamo  whose  brushes  are  not  at  the  right  position  for  spark- 
less  running. 

To  overcome  or  diminish  these  difficulties  it  is  desirable  to  avoid 
as  much  as  possible  using  alternators  driven  by  engines  whose  speed 
is  unsteady  or  irregular,  and  to  use  prime  movers  giving  a  speed  which 
is  as  uniform  and  constant  as  possible.  However,  speed-variations 
may  be  produced  in  the  best-regulated  steam  engines  and  turbines, 
by  sudden  and  large  changes  of  load.  For  this  reason  it  is  desirable 
to  apply  the  remedy  to  the  converter  itself,  even  though,  in  consequence, 
a  small  fraction  of  the  useful  energy  may  be  lost. 

This  remedy,  which  is  quite  simple,  is  well  known  since  the  "  damp- 
ers "  of  Hutin  &  Leblanc  have  been  introduced.  The  principle  in- 
volved is  that  the  speed-oscillations  themselves  cause  eddy  currents 
to  be  produced  either  in  the  pole -pieces  or  else  in  "  damping  circuits  " 
placed  suitably  around  or  about  the  pole-pieces. 

A  noticeable  damping  effect  will  already  be  obtained  by  merely 
making  the  pole-pieces  of  solid  annealed  wrought  iron.  The  damping 
effect  produced  is  not  always  sufficient,  however;  and  as  eddy  currents 
are  also  produced  in  such  pole-pieces  in  consequence  of  the  irregularity 
of  the  revolving  field  caused  by  the  two-phase  or  three-phase  currents 
employed,  the  loss  of  energy  due  to  these  eddy  currents  is  not  always 
utilized  in  improving  the  stability  of  running  of  the  converter. 

In  attempting  to  use  laminated  pole-pieces  with  the  object  of  eliminat- 
ing these  eddy  current  losses  and  of  improving  the  efficiency  of  the 
converter,  it  was  soon  found  that  the  stability  of  operation  of  the  con- 
verter was,  thereby,  made  altogether  insufficient.  The  proper  stability 
was  restored  to  the  converter  by  placing  copper  or  brass  rings  around 
the  pole-pieces,  or  else  simply  by  placing  the  field-excitation  windings 
on  copper  spools  or  forms  constituting  a  closed  electric  circuit  around 
the  field-cores  and  poles. 

These  "  damper-circuits  "  are  adjusted  in  such  a  manner  as  to 
reduce  as  much  as  possible  the  energy-loss  produced  by  the  currents 
induced  in  them,  while  at  the  same  time  obtaining  the  damping  effect 
required.  For  example,  Mr.  Parshall  has,  in  this  way,  adjusted  the 
damping  of  several  converters  of  900  KW.  capacity  by  connecting  them 
in  parallel  on  an  artificial  line.  Even  when  supplied  with  currents 
by  an  alternator  whose  speed  was  irregular  they  operated  satisfactorily 
with  a  loss  of  3  per  cent  in  the  reactances  interposed  between  them. 


222   GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

The  loss  in  the  damping  circuits  was  then  gradually  decreased  from 
3  per  cent  to  0.6  per  cent  only,  without  the  converters  falling  out  of 
step.  Experience  shows  that,  as  a  general  rule,  the  stability  of  the 
converter  is  better  when  the  supply-current  is  lagging  than  when  it  is 
leading,  probably  because  the  induced  E.M.F.  is  higher  with  leading 
currents.  Much  better  results  will  be  obtained,  both  in  respect  to  the 
damping  of  speed-oscillations  and  in  respect  to  ease  of  starting,  by 
using  the  Leblanc  dampers  in  their  most  perfect  form,  namely,  in  the 
form  of  grids  consisting  of  copper  bars  or  rods  imbedded  in  the  pole- 
pieces  (parallel  to  the  shaft)  and  connected  to  copper  plates  at  both 
ends.  (See  part  I,  Fig.  62.)  In  most  cases,  however,  to  reduce  the 
cost,  a  simpler  form  of  damper  is  used. 

USE    OF    ROTARY    CONVERTERS    FOR    TRANSFORMING    DIRECT 
INTO  ALTERNATING   CURRENT 

Since  rotary  converters  are  reversible  machines  they  can  be 
employed,  in  certain  cases,  for  producing  alternating  currents  from 
direct  current. 

There  may  be  cases  where  a  small  portion  of  the  output  of  a  direct- 
current  generating  plant  must  be  transmitted  to  a  considerable  distance. 
In  such  cases  the  power  to  be  transmitted  can  be  converted  into  high- 
tension  three-phase  alternating  currents  which,  after  being  transmitted 
to  the  receiving  end,  can  be  utilized  there  as  alternating  currents  or 
else  can  be  reconverted  into  direct  currents.  Again,  in  certain  direct- 
current  installations,  it  may  be  desirable  to  convert  a  portion 'of  the 
power  into  three-phase  alternating  currents  for  operating  certain 
apparatus,  such  as,  for  instance,  for  supplying  induction-motors,  in 
cases  where  commutator-motors  would  not  be  allowable.  It  is  also 
possible  and  it  may  be  desirable  in  certain  cases  to  produce  poly- 
phase alternating  currents  by  means  of  a  converter  supplied  with  direct 
current  from  a  storage  battery. 

In  such  cases  the  characteristics  and  the  performance  of  the  rotary  con- 
verter are  materially  different  from  what  they  are  in  the  cases  previously 
considered,  being  similar  in  all  respects  to  those  of  direct  current  motors. 

In  the  first  place  the  speed  is  not  constant,  but  it  depends  upon  the 
field-excitation,  owing  to  the  fact  that  the  E.M.F.  induced  in  the  direct- 
current  armature-winding  must  always  be  substantially  equal  to  the 
E.M.F.  of  the  source  of  (D.C.)  current,  less  the  ohmic  drop.  Hence, 
when  the  field-excitation  is  made  stronger  the  speed  decreases,  and 
vice  versa. 


SEVERAL  ROTARY  CONVERTERS  IN  PARALLEL   223 

Moreover,  the  armature-reactions  produced  by  the  reactive  current 
produce  the  same  effect  on  the  resultant  total  field -excitation.  The 
alternating  current  produced  by  the  converter,  in  such  a  case,  generally 
lags  behind  the  (A.C.)  E.M.F.,  hence  it  weakens  the  magnetic  field, 
and  this,  instead  of  causing  a  falling  off  in  voltage,  causes  an  increase 
in  speed.  If  the  A.C.  circuit  happens  to  contain  a  considerable  amount 
of  inductance,  the  reactance  will  increase  with  the  frequency  (due  to 
the  increase  of  speed);  and  the  lag  will  increase  with  the  reactance. 
The  demagnetizing  effect  will  therefore  be  increased.  This  will  cause 
a  further  increase  of  speed,  a  further  increase  of  frequency,  and  again 
a  further  increase  of  reactance;  and  the  consequence  of  this  cumulative 
process  may  be  that  the  converter  will  "  run  away  "  or  attain  an  excess- 
ive speed.  It  is  therefore  necessary,  in  such  cases,  to  take  special 
precautions,  even  including  the  use  of  safety-devices  such  as  automatic 
safety  stops,  set  so  as  to  operate  to  open  the  (D.C.)  supply- circuit 
whenever  the  converter  speed  exceeds  a  certain  limit. 

This  want  of  stability  can  be  avoided,  and  the  converter  can  be 
made  to  operate  under  synchronous  conditions,  as  before,  by  simply 
connecting  to  it  (in  parallel  with  its  A.C.  circuits),  an  alternator  which 
is  driven  at  constant  speed  by  an  engine  or  motor,  and  which  has 
sufficient  power  to  withstand  any  tendency  to  be  itself  driven  as  a  motor 
by  the  current  produced  by  the  converter.  In  that  case,  the  alterna- 
tor regulates  the  frequency,  and  it  "  sets  the  step  "  for  the  converter, 
as  it  were,  also  furnishing  to  the  converter  the  reactive  current  neces- 
sary to  bring  back  the  magnetic  field  to  the  normal  value,  whatever 
may' be  the  excitation  due  to  the  field-windings. 

Other  Special  Applications  of  Converters.  As  already  stated, 
rotary  converters  may  be  used  as  motors  for  developing  mechanical 
power  equal  to  a  poriton  of  the  electric  power  supplied  to  them  either 
on  the  A.C.  or  the  D.C.  side.  In  such  a  case,  however,  the  active 
currents  entering  and  leaving  the  converter  will  no  longer  balance  each 
other,  the  difference  between  the  energy-values  which  they  represent 
being  converted  into  mechanical  energy.  The  "  transverse"  reactions 
no  longer  offset  each  other,  and  the  heating  effects  are  increased. 

This  case  can  also  be  studied  by  the  graphical  methods  previously 
presented,  because  the  diagram  representing  the  conditions  will  be 
simply  intermediate  between  the  two  diagrams  shown  in  Figs.  27  and 
28,  corresponding,  respectively,  to  a  synchronous  motor  and  to  a 
converter.  All  that  is  needed  is  to  change  the  lengths  of  the 
segment  OB,  making  it  equal  to  a>L'(Iw—Iw')  instead  of  a>L'Iw,  in 


224   GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

which  /V  will  represent  the  reactive  current  transformed  into  direct 
current. 

It  is  also  possible  to  use  a  rotary  converter  as  a  partial  generator, 
by  applying  mechanical  power  to  its  shaft.  This  mechanical  power 
is  added  to  the  electric  power  applied  to  the  converter  from  the  electrical 
source  (A.C.  or  D.C.)  supplying  it.  The  power-output  of  the  converter, 
in  such  a  case,  will  be  equal  to  the  sum  of  the  two  power-inputs  less  the 
conversion-losses.  This  case  can  also  be  discussed  easily  by  means 
of  the  preceding  diagram. 

There  is  still  another  case,  namely,  where  a  rotary  converter  is  used 
entirely  as  a  generator)  for  generating  both  alternating  and  direct 
currents.  In  such  a  case  the  machine  is  not  operating  as  a  rotary 
converter  at  all,  but  as  a  "  double-current  "  or  "  omnibus  "  dynamo. 
The  machine  is  then  a  particular  kind  of  alternator  and  its  character- 
istics should  be  studied  as  such.  The  reactions  due  to  the  two  kinds 
of  current  no  longer  compensate,  but  add,  themselves;  hence  there 
will  be  sparking  at  the  brushes  unless  their  position  be  shifted  with 
the  load;  and  the  heating  effects  will  be  increased  instead  of  decreased, 
as  compared  with  the  operation  of  the  same  machine  as  a  rotary 
converter. 

Phase-Converters.  Instead  of  making  a  rotary  converter  produce 
direct  current  when  supplied  with  three-phase  current,  it  could  be 
made  to  supply  two-phase  alternating  current.  The  reactions  occurring 
in  such  a  machine  would  be  very  interesting,  because  they  would  be 
much  more  complicated  than  in  the  case  of  ordinary  rotary  converters. 
In  fact,  the  (two-phase)  currents  generated  by  the  machine  would 
no  longer  have  a  definite,  fixed,  phase-relation  with  the  E.M.F.,  as  in 
the  case  of  the  direct  current,  but  they  would  have  a  lag  which  would 
vary  with  the  reactance  in  the  external  circuit. 

It  could  again  be  seen  easily,  by  the  same  reasoning  as  before,  that 
since  the  power  applied  to  the  machine,  and  the  power  delivered  by 
it  have  nearly  the  same  values,  the  transverse  reactions  due  to  the 
active  current  must  practically  offset  or  balance  each  other,  leaving 
only  the  direct  reactions  due  to  the  reactive  currents.  The  latter  can 
be  evaluated  separately  and  we  can  thus  obtain,  for  each  value  of  the 
power  output  and  of  the  lag  in  the  secondary  circuit,  the  resultant 
magnetomotive  force  produced  by  the  reaction  of  the  armature  on  the 
magnetic  circuit. 


CHAPTER  V 

VOLTAGE  RATIO  IN  SYNCHRONOUS  CONVERTERS  WITH  SPECIAL 
REFERENCE  TO   THE  SPLIT-POLE  CONVERTER 

Resume  of  paper  by  C.  A.  ADAMS,  Proc.  A.I.E.E.,  June,  1908 

THE  author  of  this  paper  shows  how  the  ratio  of  the  B.C.  to 
the  A.C.  voltage  of  a  synchronous  converter  may  be  varied  by 
means  of  the  split  pole,  without  introducing  seriously  large  harmonics 
into  the  counter  E.M.F.  and  current.  Incidentally  a  new  method 
is  employed  for  calculating  the  E.M.F.  between  adjacent  commu- 
tator brushes — the  brush  E.M.F. — and  the  corresponding  E.M.F. 
between  collector  rings — the  tap  E.M.F. — for  any  given  flux  dis- 
tribution. 

The  elementary  E.M.F.  induced  in  a  single  conductor  cutting 
through  the  gap-flux  at  constant  speed  is  proportional  to  the  density 
of  the  flux,  and  when  plotted,  will  obviously  have  the  same  shape  as 
that  of  the  gap-flux  curve.  It  may  be  expressed  thus: 

sin  to/ + gas  sin  30)/+ga5  sin 
+qam  sin 

,1  COS  (i>/+<7&3  COS 


-\-qbm  COS  WW/+etC.  (l) 

where  a>  =  2i:n  and  n_  is  the  fundamental  frequency. 

The  brush  voltage  which  is  proportional  to  the  area  of  the  e'  curve 
between  brushes,  is 

.     .     (2) 


or 


where  N  is  the  number  of  conductors  in  series  between  brushes. 

The  tap  E.M.F.  is  the  geometrical  sum  of  the  E.M.F.'s  of  the 
several  coils  connected  in  series  between  taps.    The  E.M.F.  of  each 

225 


226   GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

coil  is  in  turn  the  geometrical  sum  of  the  E.M.F.'s  of  its  two  sides, 
which  E.M.F.'s  may  or  may  not  be  in  phase  according  to  whether  or 
not  the  coil  pitch  is  equal  to  the  pole  pitch.  The  fundamental 
E.M.F.'s  of  the  two  sides  of  each  coil  will  differ  in  phase  by  the 
deficiency  in  coil  pitch  measured  in  electrical  degrees,  and  the  funda- 
mental E.M.F.'s  of  the  coil  sides  in  two  adjacent  slots  will  differ  in 
phase  by  the  electrical  angular  pitch  of  the  slots.  In  each  case  the 
phase  difference  between  the  mth  harmonics  is  m  times  that  between 
the  fundamentals. 

In  calculating  the  tap  E.M.F.  the  author  introduces  the  "  differ- 
i  V/j  ^jntial  factor,"  the  ratio  of  the  resultant  or  geometrical  sum  to  the 
arithmetrical  sum  of  the  various  E.M.F.'s. 

^f  Let  Nji^=_  slots  per  pole,  and  p'  the  number  of  belt  spans  per 
X  electrical  circumference  (pr  =  2  for  diametral  connection  and  3  for 


three-phase  delta  or  six-phase  double  delta);    then  — —  =the  phase 


difference  between  the  fundamental  E.M.F.'s  of  two  adjacent  slots, 
—  —  =  the  phase  difference  between  the  corresponding  mth  harmonics 

J\  sp 

and  —^-  =  slots  per  belt.     The  total  phase  rotation  (or  combined 

phase  displacement)  of  the  mih  harmonic  in  all  the  slots  of  the  belt 

will  be 

mit      2Nsp 


and  the  resultant  of  the  — 7^  mth  harmonics  is  proportional  to 
2  sin  —7-.  By  the  same  constant  a  single  mth  harmonic  is  proportional 
to  2  sin  -jrr~»  and  the  arithmetical  sum  of  the  — -~  wth  harmonics  is 

21\sp  p 

2NsP^,      .      niT 
—-X2  sin 


2Nsp 


The  differential  factor  for  the  mth  harmonic  of  the  belt  E.M.F.  ; 
or  the  belt-differential  factor  for  the  mth  harmonic  is  then 


.    mit 


sin  —  IT— 

2J\sp 


VOLTAGE  RATIO  IN  SYNCHRONOUS  CONVERTERS     227 

and  that  for  the  fundamental 

.    iu 

P'    S1V 


2Nsp  1C 

sm  — ry- 

2Nsi 


(s) 


The  ratio  kdm+kai  is  the  relative  reduction  of  the  mth  harmonic 
by  belt-differential  action;  it  may  conveniently  be  called  the  belt- 
reduction  factor,  and  is 


, 

kdm 


sin 


sin 


.    x       .      WIT: 
sin  -7    sin  — — 

f  2Nsp 


(6) 


The  first  part  of  krm,  namely, 


,  is  independent  of  Nsp  and  is 


equal  to  i,  except  for  w  =  3,  9  or  15,  when  it  is  zero  for  p'=$)  and 
2  for  p'  =  6. 

The  following  table  gives  the  value  of  the  second  part  of  kTm, 
which  is  equal  to  krm  except  for  the  cases  above  cited. 


TABLE  II 


TABLE   OF   sin—      -  sin      7— 

2Nsp  2Nsp 


»5p= 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

00 

m=  3 

o.35i 

0-344 

0.340 

0.338 

0.3368 

0.3362 

0.3358 

0-3355 

0.3353 

0.3351 

T 

m=  5 

0.235 

O.  222 

0.214 

O.2IOI 

0.2081 

0.2062 

0.2051 

o.  2042 

0.2034 

o.  2029 

JL_ 

m=  7 

0.  199 

0.  1755 

o.  1646 

o.  158 

0.154 

0.152 

0.150 

0.149 

o.  1477 

0.1467 

\ 

m  =  9 

o.  198 

o.  158 

o.  141 

0.132 

o.  126 

0.123 

o.  1208 

o.  119 

0.1177 

0.1166 

•g 

W  =  II 

0.2345 

0.1583 

o.  1316 

o.  1185 

o.  mi 

o.  106 

o.  103 

o.  1009 

0.0992 

0.0979 

y1^ 

W  =  13 

0.351 

0.1755 

o.  I3i7'o.  1124 

O.  IO2 

0.961 

0.0920 

0.08916 

0.08702 

0.08533 

yV 

tn  =  i5 

0.995 

0.2214 

o.  1413  o.  1126 

0.0986 

o  .  0903 

0.08492 

0.0813 

0.07865 

0.07665 

yV 

I  .OOO 

o  .  3444 

0.1644 

o.  1185 

0.0985 

0.0875 

0.0807 

0.0761 

0.0729 

0.0706 

yV 

m  =  19 

0.351 

I.  OOO 

0.214 

o.  132 

o.ioo  (0.0875 

0.0788 

0.073 

0.069 

o  .  0662 

X 

As  krm  is  in  most  cases  much  less  than  one,  it  appears  that  the 
harmonics  are  much  more  reduced  by  the  belt-differential  action  than 
is  the  fundamental. 


228    GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

In  the  case  of  a  fractional  pitch  or  short-chord  winding,  there  is  a 
phase  difference  between  the  E.M.F.'s  in  the  two  sides  of  each  coil, 
or  in  the  ordinary  two  layer  winding,  this  same  phase  difference 
exists  between  the  E.M.F.'s  of  the  two  belts  or  groups  of  coil  sides, 
which  together  make  up  one  circuit  between  taps.  This  introduces 
another  reduction  of  the  resultant  E.M.F.  If  Ac  is  the  coil  pitch  in 
terms  of  full  pitch,  the  pitch  differential  factor  between  the  two  belts 

is,    for   the   fundamental   (see    Fig.    29),  OB+(dA+OB)  =  sm  -Xc, 

and  for  the  rath  harmonic  is  sin  — AC.      The  ratio  of  the  latter  to  the 

2 

former  will  be  designated  the  pitch  reduction  factor  for  the  rath  har- 
monic; it  is 


sin  ra— Xc 

2      _  per  cent,  of  wth  harmonic  in  fractional  pitch  winding         /  >. 

,    x  per  cent,  of  mth  harmonic  in  full  pitch  winding        *       *'' 

sm  —Ac 
2 


This  is  plotted  in  Fig.  30  for  various  pitches  and  harmonics 
(signs  disregarded). 

If  the  phase  belt  begins  and  ends  in  the  middle  or  within  a  coil 
or  slot  bundle,  there  is  a  still  farther  reduction  of  the  resultant 
E.M.F.  The  corresponding  fractional  slot  reduction  factor  is 


ksm  — 


cos  -^—  (for  half-slot  connection). 

rax  rax    (for   a   one-third   or   a   two- 
cos  — r-  COS  ,  .  .         • 

2JMsp  oNsp      thirds  slot  connection). 

rax  rax    (for  a  one-fourth  or  a  three- 

cos -^ — '-cos-=-  ^   ,        .      . 

2Nsp          ^Nsp      fourths  slot  connection). 


-     (8) 


VOLTAGE  RATIO  IN  SYNCHRONOUS  CONVERTERS    229 

The  final  ratio  of  the  per  cent  mth  harmonic  in  the  phase  E.M.F. 
to  the  per  cent  mth  harmonic  in  the  flux  distribution  curve  is 


.      WX.  .      WX         .          X 

sin  — Ac    sin  — T    sm  — :-r 


krmXkpmXksm^ 


___ 

X "X 

.X.  .      X  .        WX 

sin  —Ac      sin  -7     sin  — -— 

2  p'  2Nsp 


•     (9) 


It  remains  to  consider  the  phases  of  the  harmonics  in  the  resultant 
E.M.F. 


.     2X 


The  total  phase  rotation  of  the  fundamental  is  -7,  and  the  phase 

P 


//    V 


.82  .84  .86  .88  .90 

Coil  Pitch  in  Terms  of  Full  Pitch 

FlG.   30. 


of  the  fundamental  of  the  resultant  E.M.F.  with  respect  to  the 
fundamental  of  slot  No.  i  is 


'       2Nsp 


2WX 


The  total  phase  rotation  of  the  mth  harmonic  is  — r. 

P 


m 


Let  w= nearest  whole  number  less  than  -7  and  ws=  nearest  whole 

/yyt 

number  less  than  — =-.     Then  the  equivalent  phase  rotation  of  the 

21\  sp 


230   GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

(  \ 

— —  w  j ,  and  the  phase  of  the  mth  harmonic  of 

the  resultant  E.M.F.  with  respect  to  the  mih  harmonic  of  slot  No.  i 
is,  in  mih  harmonic  radians, 


(m        \        /    m 
-7-031-xl^^- 


1   '      Belt  Spah  in  Degrees 


FIG.  31. 


Thus  in  passing  from  the  slot  E.M.F.  to  the  resultant  E.M.F., 
the  relative  phase  of  the  mih  harmonic  with  respect  to  the  funda- 
mental is  changed  by  an  amount  (in  mih  harmonic  radians), 


—  1C  (to  —  tos). 


P  is  therefore  always  a  multiple  of  x.  Moreover,  W*  is  practically 
always  zero.  Therefore,  $m  is  zero  when  w  is  even  and  x  when  w 
is  odd;  i.e.,  the  wth  harmonic  can  only  be  reversed  and  not  other- 
wise changed  in  phase  from  the  slot  E.M.F.  In  order  to  see  this 
relation  more  clearly,  consider  an  infinite  number  of  slots,  and 


VOLTAGE  RATIO  IN  SYNCHRONOUS   CONVERTERS    231 

designate  the  angle  of  belt  span  by  cj>.     The  belt  reduction  factor 
is  then 


.    cj> 


2 


which  is  plotted  in  Fig.  31  for  different  values  of  the  belt  span  (cj>) 
and  for  several  harmonics.  The  reversal  of  the  various  harmonics 
with  the  change  in  belt  span  is  here  shown  clearly. 


TAP  VOLTAGE 
For  this  purpose  Eq.  (i)  may  be  written: 


e  =  aiqi  sn    (•>        i£3  sn 

+#5  sin  (sw/+05)  +  .  .  .  qm  sin  (ww/+0m)+etc.]      .     (n) 
where 


_ 

qm=  \/qam2  -\-qbm2     and     0m=  tan"1  —  . 

am 


The  mih  harmonic  in  the  tap  E.M.F.  is  then 

kmqm  sin  (f»&>H-6»-Hc<i>). 
The  amplitude  of  the  fundamental  is 

2N 

ati  =  kdi—rq\a\1 


p'   .    TU 
-sin-7 
x        p 

Eq.  (5),  as  Nsp  approaches  infinity.     Then 


where  kd\  may  be  taken  as  equal  to  —  sin  ^7  the  limiting  value  of 


-Naiqism-;=qiEi  sin -7,     ....     (12) 


232  GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 
and  the  complete  tap  voltage  is 

et=qiEi  sin—  ,[sin  ut+kzqz  sin  (3G>/+03+x(i>) 

+k5q&  sin  (50)*+  85+™)+  etc.]     .     (13) 


The  root  mean  square  tap  voltage  is  then 


sn 


, 

V2  P 

and  the  ratio  of  brush  to  tap  voltage  is 


.,  .     .     (14) 


Edc         v  2              yj.-r?y3T'(f¥«&T-T¥t*y  r  ^^-  •  -  •/         /     \ 
jfV{,f=  — —  = —          — /\ — .  ^=,      v1^/ 

gi  sin  ^- 

'  /       • 
K* — ^  •  •  • / (I6) 

81  «n^ 

when 

d7) 


THREE-PART  POLE 

With  a  normal  flux  distribution,  or  with  a  three-part  pole  sym- 
metrical distortion,  the  qb's  are  zero,  91  =  1  and  qm  =  qam.  The  denomi- 
nator of  Kq  is  nearly  unity,  always  a  little  greater.  The  numerator 
may  differ  considerably  from  unity  on  either  side,  according  as  the 
qs's  are  positive  or  negative,  slightly  less  than  unity  for  undistorted 
flux  distribution.  Thus  Kbt  will  be  slightly  less  than  as  given  by  the 
usual  approximate  formula.  Any  variation  in  the  £a's,  due  to  a  sym- 
metrical distortion,  causes  a  variation  in  the  voltage  ratio,  but  also 
in  the  harmonics  of  the  tap  E.M.F.'s. 

The  relation  between  these  two  changes  is  shown  in  Fig.  32  for 
the  180°  E.M.F.,  and  for  the  third,  seventh,  eleventh,  and  seventeenth 
harmonics,  each  being  taken  separately. 


VOLTAGE  RATIO  IN  SYNCHRONOUS  CONVERTERS    233 

In  order  to  change  the  relative  voltage  ratio  from  i  to  .9  by  third 
harmonic  control,  involves  a  10  per  cent  third  harmonic  in  the 
diametral  tap  E.M.F.,  which  corresponds  to  about  30  per  cent  third 
harmonic  in  flux  distribution  curve.  To  make  the  same  change  by 
seventeenth  harmonic  control,  involves  a  12  per  cent  seventeenth 


11 


n 


1.8 


1.2 


Km 


17 


i  •?    -f    • 

Qm=\%  Harmoriic  in 


180 


FIG.  32. 

harmonic  in  diametral  tap  E.M.F.,  and  a  seventeenth  harmonic  in 

12 


the  flux  distribution  curve  of  7 —  per  cent. 

kr17 


For  twenty  slots  per 


pole,  this  would  mean  a  150  per  cent  seventeenth  flux  harmonic.  A 
combination  of  harmonics  produces  about  the  same  result  as  a  single 
one  of  the  same  total  per  cent  magnitude. 


234   GENERAL  DIAGRAMS  FOR  SYNCHRONOUS  MOTORS 

The  use  of  fractional  pitch  windings  and  fractional-slot  connec- 
tions, will  reduce  some  of  the  harmonics;  but  as  commutation  limits 
the  pitch  reduction  to  a  comparatively  small  range,  only  the  higher 
harmonics  are  affected. 

If  the  voltage  ratio  could  be  under  control  by  a  pure  third  har- 
monic in  the  flux  distribution  curve,  the  resulting  third  harmonic  in 
the  phase  E.M.F.  could  be  eliminated  from  the  line  voltage  by  con- 
necting the  transformer  primaries  in  star. 


TWO-PART  POLE 

This  gives  unsymmetrical  distortion,  and  a  lateral  shifting  of  the 
flux  distribution  curve;    i.e.,  the  q^s  are  no  longer  zero,  particularly 


FIG.  33. 

qb\.  Thus  qi  —  \/i-\-qbi2  is  no  longer  unity,  and  Kbt  is  changed  by  qi 
as  well  as  by  Ka.  If  these  changes  are  in  the  same  direction,  the 
harmonics  in  the  tap  E.M.F.  will  be  obviously  less  for  a  given  dis- 
tortion. 

In  an  actual  machine  for  which  the  flux  distribution  curve  is 
shown  in  Fig.  33,  the  voltage  was 


Edc 
Et 


i8oc 


—       (I±J?«3+-k'*5+etc.)     =  A/2   0.862        /j 
9i     Vi+to+fa+etc.     J-295  1.016 


less  than  two-thirds  of  the  ratio  for  a  sinusoidal  flux  distribution 
curve,  and  just  about  two-thirds  of  that  for  a  normal  undistorted 


VOLTAGE  RATIO  IN  SYNCHRONOUS  CONVERTERS    235 

field.  The  total  harmonics  were  21.6  per  cent  in  the  180°  E.M.F., 
and  only  7.2  per  cent  in  the  120°  E.M.F.,  which  are  considerably 
less  than  is  possible  with  a  three-part  pole.  Moreover,  it  is  quite 
possible  that  a  more  careful  design  of  the  pole  faces  might  result  in 
a  farther  reduction  in  the  harmonics  of  the  tap  E.M.F.'s.  Fractional 
pitch  winding  and  fractional  slot  connections,  would  reduce  them 
still  slightly  farther. 


PART  III 

METHODS  OF  CALCULATION  OF  THE  ARMATURE  REAC- 
TIONS (DIRECT  AND  TRANSVERSE)  OF  ALTERNATORS 


CHAPTER  I 

METHODS    OF  CALCULATION    OF  THE    ARMATURE    REACTIONS 
(DIRECT  AND  TRANSVERSE)  OF  ALTERNATORS  l 

THE  author  here  proposes  to  explain  and  complete  the  theory  of 
"  two  armature  reactions,"  which  was  enunciated  by  him  several 
years  ago,2  and  which  has  recently  been  adopted,  with  slight  modi- 
fications, by  M.  Key,3  M.  R.  V.  Picou,4  and  M.  Guilbert,5  in  France; 
Professor  Arnold  6  in  Germany;  and  Mr.  Herdt7  and  Messrs.  Hobart 
and  Punga  8  in  the  United  States.  The  notable  authority  of  all  these 

*A  paper  by  Prof.  Andre  Blondel,  Ecole  des' Fonts  et  Chausees,  presented 
before  the  International  Electrical  Congress  at  St.  Louis,  in  1904.  Reprinted 
from  the  Transactions,  Vol.  I,  pp.  635-668. 

2  "On  the  empirical  theory  of  alternators,  "L' Industrie  Electrique,  Nov.  lo 
and  25,  1899.     This  is  the  first  publication  in  which  the  reaction  in  alternators 
was  analyzed,  and  possesses  undisputable  priority  over  all  those  which  are  men- 
tioned below  on  the  subject  of  the  two  reactions. 

3  M.  Rey.     Rapports,  International  Congress  of  Electricians,  1900. 

4  M.   R.    V.    Picou.     Bulletin    de  la    Societe    Internationale    des   Electriciens 
July,  1902. 

*  C.  F.  Guilbert.  Eclairage  Electrique,  March  7  and  14  and  April,  1903,  and 
La  Revue  Technique,  June,  1903. 

6  E.  Arnold.     Elek.  Zeit.,  1902,  page  250.     Arnold,  as  pointed  out  farther  on, 
has  reduced  the  generality  of  the  method,  in  contradistinction  to  the  other  authors 
mentioned. 

7  L.  A.  Herdt.     Trans.  Amer.  Inst.  El.  Eng.,  May,  1902,  and  Eclairage  Elec- 
trique, February  14,  1903. 

8  Hobart  and  Punga.     Trans.  Amer.  Inst.  El.  Eng.,  April  22,  1904. 

236 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS       237 

authors  in  the  matter  of  dynamo-machine  construction  has  made  me 
read  their  communications  with  great  interest,  and  as  I  have  observed 
that  in  certain  cases  my  own  view  has  not  been  well  understood,  I 
consider  it  desirable  to  present  certain  supplementary  considerations 
to  make  this  theory  still  more  simple  and  to  complete  it  finally.  At  the 
outset  it  should  be  pointed  out  that  my  diagram  should  not  be  con- 
sidered as  belonging  to  the  category  of  E.M.F.  but  rather  to  that 
of  ampere-turn  diagrams.  The  two  classes  are  often  equivalent,  because 
if  one  commences  with  E.M.F. 's  one  proceeds  with  fluxes,  and  ends 
necessarily  with  ampere-turns.  But  I  desire  to  reduce  to  a  minimum 
the  complication  of  considerations  relative  to  the  saturation  of  field- 
magnets,  of  which  I  fear  the  difficulties  have  been  needlessly  exaggerated. 
In  what  follows  I  will  refer  first,  very  briefly,  to  the  essential 
points  of  my  method  of  1899,  and  I  will  show  in  what  points  it  has 
been  improved,  or  is  susceptible  of  improvement. 

PART  I.    DIAGRAM  OF  OPERATION. 

Principles  of  the  Theory  of  Two  Reactions.  I  have  long  been  sur- 
prised that  polyphase  alternators  and  direct-current  machines  have 
not  been  treated  from  the  point  of  view  of  reaction,  since  these 
phenomena  are  fundamentally  of  absolutely  the  same  order,  since 
the  dephasing  alternating  current  produces  effects  of  the  same  order 
as  the  displacing  of  the  brushes  in  a  direct-current  dynamo.  It  is 
known  that  in  the  latter  case  the  displacement  causes  a  direct  magnetic 
reaction  to  be  developed,  whilst  in  the  neutral  position  there  is  only 
a  transverse  reaction.  By  similar  reasoning  as  to  the  automatic 
dephasing  of  alternating  currents  and  of  the  property  which  polyphase 
currents  possess  of  being  decomposable  into  active  and  reactive .  com- 
ponents, I  have  been  led  to  the  following  proposition: 

When  an  alternator  supplies  a  current  dephased  by  an  angle  <I>  with 
respect  to  the  internal  induced  E.M.F.,  the  armature  reaction  may  be 
considered  as  the  resultant  of  a  direct  reaction  produced  by  the  reactive 
current  sin  (f>  and  a  transverse  reaction  due  to  the  active  current  I  cos  $. 

In  addition  to  the  above,  the  stray  magnetic  fields  must  be  taken 
into  account.  These  are  proportionate  to  the  currents  and  in  phase 
with  them.  We  will  consider  them  later  on. 

The  second  fundamental  proposition  of  this  theory  is  the  follow- 
ing: 

The  two  reactions  (direct  and  transverse)  and  the  stray  flux  take 


238  METHODS  OF  CALCULATION 

place  in  three  different  magnetic  paths;  only  the  direct  reaction  acts  in 
the  main  circuit  of  the  field  magnets,  while  the  transverse  reaction  and 
the  stray  fields  act,  in  general,  upon  circuits  of  low  magnetic  density. 

The  conclusion  which  I  have  drawn  from  the  above  is  that,  in 
general,  the  direct  reaction  should  be  expressed  as  a  counter  M.M.F.; 
that  is  to  say,  by  a  number  of  ampere-turns  equivalent  to  the  demag- 
netizing effect  of  the  armature.1  The  total  armature  M.M.F.  per 
complete  magnetic  circuit  or  per  pair  of  poles  is 

KNT  /- 

--IV,, 

where  K  is  a  coefficient  of  reduction  and  N  the  armature  conductors 
per  pair  of  poles.  I  have  formally  given  for  asynchronous  motors  a 
practical  value  which  is  approximately  the  same  for  alternators,  viz.: 

'    -©V. 

k  being  the  coefficient  of  reduction  which  appears  in  the  formula  of 
E.M.F.  written  under  the  form 


Here  N  is  the  number  of  peripheral  conductors  for  one  phase  and  cu  the 
velocity  of  pulsation.  It  is  the  direct  reaction  which  produces,  almost 
entirely,  the  variation  of  terminal  voltage.  As  to  the  transverse  reac- 
tion and  the  reaction  of  stray  fields,  with  the  assumption  that  the 
armature  is  unsaturated,  as  I  assumed  and  as  M.  Guilbert  also 
assumes,  they  may  be  expressed  simply  by  the  coefficients  of  self- 
induction  /  and  s. 

More  recently  I  have  indicated  2  that  the  transverse  reaction  could 
also  easily  be  expressed  in  ampere-turns. 

The  analysis  of  the  phenomena  taking  place  in  the  alternator  leads 
therefore  to  a  new  proposition,  formulated  in  my  articles  of  1899. 

1  No  notice  is  here  taken  of  one  of  the  cases  considered  by  the  author  in  1899; 
.namely,  that  in  which  all  the  machine  is  well  below  saturation,  because  it  is  only 

susceptible  of  very  rare  applications;  moreover,  it  has  been  treated  with  more 
detail  by  M.  Jean  Rey  in  a  very  interesting  communication  presented  to  the 
Congress  of  1900,  in  which  the  reader  will  find  an  interesting  example  of  a  cal- 
culation of  reactive  coefficients  in  a  machine  actually  built  by  this  method,  which 
has  since  been  followed  by  various  authors. 

2  "Theory  of  Synchronous  Motors,"  Vol.  I.     Paris,  Gauthiers-Villars,  1900. 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS       239 

The  df  phasing  $  oj  the  current  is  regulated  entirely  by  the  numerical 
value  of  the  transverse  reaction,  which,  on  the  contrary,  has  little  effect 
upon  the  E.M.F. 

This  proposition  has  been  demonstrated  in  the  case  of  unsatu- 
rated  armatures,  as  I  have  just  stated,  but  it  is  general  and  remains 
in  effect  even  in  the  case  where  the  circuit  of  the  transverse  reaction 
approaches  saturation.  The  demonstration  of  this  will  be  given  below. 

Diagram  of  E.M.F. 's  and  Currents  of  an  Alternator  with  Unsat- 
urated  Armature  and  with  Saturated  Field  Magnet.  The  diagram 
in  Fig.  i  !  reproduces  Fig.  5  of  my  first  paper,  supplemented  by  the 


definitions  of  Fig.  2.  r1  represents  the  apparent  resistance,  that  is 
to  say,  the  ohmic  resistance  augmented  by  the  effects  due  to  Foucault 
currents;  (f>  is  the  difference  of  phase  in  the  external  circuit,  and  </> 
is  the  difference  of  phase  with  respect  to  the  internal  E.M.F.  It  is 
proposed  to  calculate  the  excitation  necessary  to  develop  an  E.M.F. 
U  at  the  terminals  under  a  current-delivery  7  dephased  by  0.  We 
have  OP=r'I  and  PA  =  U,  with  the  angle  APa=(f>. 

Let  Or  be  the  direction,  as  yet  unknown,  of  the  internal  E.M.F. 
e\  the  perpendicular  AB  let  fall  from  A  upon  O T  is  the  sum  of  the 
transverse  reaction: 

AG=a>lI  cos  <j>, 

1  The  subscript  w  indicates  the  active  or  energy  component  and  the  sub- 
script d  the  reactive  or  quadrature  reactive  component. 


240  METHODS  OF  CALCULATION 

(where  /  is  the  transverse  self  -inductance  and  &»(  =  2x/)  the  speed  of 
pulsation);  and  of  a  part  of  the  stray  field  reaction  — 

GB=a>sI  cos  <f>l= 

where  s  is  the  self-induction  of  the  stray  fields. 

The  segment  GD  perpendicular  to  7    represents  the    E.M.F.  of 
reaction  of  the  stray  fields,  wsl,  and  the  segment 


sn  (  = 

represents  the  second  component  of  the  stray-field  reaction;  thus  in 
OD  is  obtained  the  value  of  the  effective  E.M.F.  e,  which  should  be 
obtained  by  the  resultant  excitation. 

The  value  of  the  angle  0  is  determined  by  expressing  simply  the 
relations  between  the  elements  of  the  figure.  Let  us  analyze  the 
broken  line  OPAB  into  components  upon  OB,  and  BA\  whence 

e=r'I  cos  ^+  Ul  sin  (^-</>) 

a>(l+s)I  sin  (f)  =  rfl  sin  <p  +  U  sin  (^'  —  $) 
and 


The  angle  of  real  dephasing  0  is  thus  determined  solely  by  a 
knowledge  of  the  transverse  reaction.  This  equation,  which  was 
given  by  the  author  in  1899,  ^s  evidently  equivalent  to  the  following 
construction. 

From  the  point  A  a  perpendicular  AH  is  drawn  to  the  direction 
of  the  current  /,  and  a  segment  AF  is  drawn  upon  this  line  equal 
to  ajsl;  then  a  segment  FT=aJlI\  finally  the  point  O  is  joined  to 
the  point  Tt  and  thus  is  obtained  the  angle  0  and  the  position  of  the 
required  vector  OD  representing  the  total  effective  E.M.F.  e.  To 
determine  the  necessary  ampere-turns  for  the  production  of  this  E.M.F., 
it  is  only  necessary  to  employ  the  open  circuit  characteristic  or 
saturation  curve  of  the  alternator. 

1  The  segment  BT  intercepted  by  OT  will  evidently  be  equal  to 
BD+  DT=  toll  sin  ^+  us  I  sin  <p. 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS       241 

Diagram  of  Ampere-Turns  in  the  Case  of  Unsaturated  Armature. 

The  consideration  of  ampere-turns  does  not  need  to  appear  in  the 
method,  as  is  evident  in  the  case  of  an  unsaturated  armature,  until 
after  having  traced  the  diagram  of  E.M.F/s.  The  excitation  ampere- 
turns  are  drawn,  if  desired,  along  the  direction  of  the  vector  OC, 
in  order  to  facilitate  certain  comparisons;  but  the  calculation  of  ampere- 
turns  is  no  longer  in  this  method  a  vectorial,  geometric  calculation,  but 
a  scalar  calculation,  and  may  be  made  upon  the  saturation  curve  as 
calculated  or  observed,  which  represents  the  induced  E.M.F.  (or  the 
useful  flux  traversing  the  armature)  as  a  function  of  the  total  ampere- 
turns  applied  on  the  field  magnets. 

Their  determination  is  based  on  the  following  facts: 
(i)  The  reactive    counter-ampere-turns  of  the  armature  are  pro- 
portional to  the  number  of  peripheral  conductors  on  the  armature  per 

pair  of  poles,  N2,  which  supply  a  number  of  turns  equal  to  —  -;    but 

these  turns  do  not  act  in  unison,  partly  because  they  belong  to  different 
phases,  and  partly  because  they  are  not  in  the  same  slots.  For  this 
reason  it  is  necessary  to  apply  a  reduction-coefficient  K.  The  effect- 
ive reactive  current  I2  sin  <p  thus  gives  rise  to  an  M.M.F.  exactly 
opposite  to  that  of  the  field  magnets,  and  having  for  its  magnitude 
in  counter-ampere-turns 


sin  <[>  KN2Id 


(2)  These  counter  ampere-turns  act,  on  the  one  hand,  upon  the 
circuit  common  to  the  field  magnets  and  the  armature;  and  on  the 
other  hand,  upon  the  circuit  of  the  arma-  .,  T 

T>  "* i¥1  J.1  .— 

ture  and  of  the  stray  magnetic  fields.     This 
may  be  represented  diagrammatically  as  in 


Fig.  3,  in  which  the  full  lines  ABCDEF 
represent  the  principal  magnetic  circuit  with 
its  two  M.M.F.'s  acting  in  the  opposite 


KN2It 


Field 


Field  leakage 


ArmatureJeakage/2 


D 

fl  Air  Gap 


directions  N\Ii  and =.— ,  and  the  dotted 

V2  A 

lines  AD  AD  indicate  the  stray  circuits  in       | 

which  the  lines  of  force  escape,  either  be-  ~*&a     & 

tween  the  teeth  of  the  armature  /2  or  between  FIG.  3. 

the  polar  horns  t\.     In  reality  the  stray  fields 

/i  of  the  field  magnet  are  not  concentrated  along  any  single  path,  but 


242  METHODS  OF  CALCULATION 

are  spread  out  more  or  less  over  the  entire  length  of  the  principal 
circuit  up  to  its  entrance  into  the  armature  turns.  This  fact  is,  how- 
ever, unimportant,  as  has  recently  been  shown  by  M.  Guilbert  (see 
Eclair  age  Electrique,  December,  1903). 

(3)  The  self  -inductance  of  the  armature  is  produced  by  the  stray 
fields  /2,  supposedly  attributed  to  the  effect  of  the  armature.  If  we 
call  R/2  the  reluctance  of  the  circuit  of  the  stray  fields  /2,  and  Ra  the 
reluctance  of  the  armature,  the  stray  field  produced  by  the  armature 
across  itself,  is  expressed  in  practical  units  (N2  being  the  number  of 
peripheral  conductors  per  pair  of  poles), 


sn  <f        o.2n222  sn 


assuming  that  we  can  neglect  Ra  with  respect  to  R/.,}  and  thus  pro- 
duce an  E.M.F. 


2\/2 


It  results  from  this  that  the  E.M.F.  of  self-induction  that  we  have 
called  wsld  can  be  considered  as  produced  simply  by  a  stray  field 
/2,  which  is  added  to  the  stray  field  of  the  field  magnets  /i. 

Upon  the  saturation  curve  XM  (Fig.  2)  defined  as  above,  the 
point  b  which  corresponds  to  the  E.M.F.  OB  (Fig.  i),  represents  the 
NI  necessary  to  force  the  useful  flux  through  the  field  magnets  into 
the  armature.  Adding  to  the  flux  4>a  the  stray  field  of  the  armature 
/2,  there  is  obtained  the  virtual  E.M.F.  OD=e,  corresponding  to 
the  total  flux  emanating  from  the  poles  into  the  air-gap;  the  corre- 
sponding abscissa  XQf  represents  the  necessary  field-winding  ampere  - 
turns  Nili',  without  taking  into  account  the  increase  AJi  of  the 
stray  field  /i  of  the  field  magnet. 

(4)  The    stray  field  of    the    field  magnets  f\  is  inversely  propor- 
tional to  the    reluctance  of  the  stray  path  Rfv  between  the  poles  and 
directly  to    the  difference  of  magnetic  potential    between  the  poles. 
This  latter  is  formed  of   two  parts;    one  part  is  the  drop  of  mag- 
netic potential  necessary  to  force  the  flux  through  the  armature  and 
air-gap,  the    other    part,  the    reactive    counter-ampere-turns    of    the 
armature  calculated  as  above. 

(5)  Every  increase  in  the  ampere-turns  of  the  field  magnet  increases 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS        243 

the  stray  flux/!  of  the  field  magnet,  in  a  manner  sensibly  proportional 
to  the  increase  of  the  ampere-turns  of  the  field.  If,  therefore,  the 
field-magnet  ampere-turns  are  increased  by 


in  order  to  compensate  for  the  counter-ampere-turns,  CAT,  of  the 
armature,  the  stray  flux  /i  produced  by  the  field  magnet  following 
the  circuit  BADC  would  be  increased  by  a  quantity, 

(CAT] 


Rl  being  the  reluctance  of  the  field  magnet. 

This  increase  of  the  flux  through  the  field  magnet  increases  the 
magnetic  density  in  the  latter  and  demands  consequently  a  correction, 
as  I  pointed  out  in  1899,  without  tracing  it  in  detail.  At  that  time 
I  conducted  the  inquiry  simply  as  follows,  supposing  the  role  of  the 
field  magnets  to  be  sufficiently  unimportant  to  permit  approximate 
correction  being  applied. 

Let  BI  be  the  flux  density  in  the  field  magnet,  corresponding  to 
the  no-load  E.M.F.  e,  that  is  to  say,  to  the  flux  </>a+/i;  and  let  us 

call   ^i    the    Hopkinson    coefficient        .  —  .     The   full-load   induction 

</>a 

will  be 

approximate,y 

(CAT)/,      } 


and  consequently  the  total  ampere-turns  will  be  increased  by  the 
quantity  by  which  the  change  from  BI  to  BI  increases  the  ampere- 
turns  (which  we  shall  call  A^/i)  specially  absorbed  by  the  reluc- 
tance of  the  field  magnets  in  the  condition  considered.  The  point 
d'  upon  the  curve  will  be  in  consequence  displaced  toward  the  left  by 
the  quantity  corresponding  to  this  increase  of  ampere-turns. 

Starting  from  the  point  Qf  duly  corrected,  it  is  sufficient  to  take 
a  length  representing  the  ampere-turns  equal  to  CA  T  of  the  armature 


244 


METHODS  OF  CALCULATION 


in  order  to  obtain  the  total  necessary  ampere-turns  OQ".  Fig.  4 
shows  how  the  diagrams  of  E.M.F/s  and  of  ampere-turns  may  be 
united  upon  a  single  sheet. 

The  preceding  reasoning  may  be  summed  up  in  the  following 
simple  equations,  employing  ordinary  language: 

The  fall  of  magnetic  potential  in  the  armature  and  air  gap  is  a 
function  of  the  flux  utilized  in  the  armature  <£a  +  the  armature  stray 
flux/2. 

The  magnetic  difference  of  potential  between  the  pole  pieces  =  the 
fall  of  magnetic  potential  in  the  armature  and  air-gap + the  reactive 
counter  ampere-turns  of  the  armature. 


Q"  Q'       Q  X  O          Jd 

FIG.  4. 

The  magnetic  stray  field  between  the  magnets  f\+^f\  =  a  function 
of  the  difference  of  potential  of  the  pole  pieces. 

The    total   stray   magnetic   fluxes  =  the    fluxes  /i+J/i+the    stray 


The  total  flux  in  the  field  magnet  =  the  useful  flux + the  flux  of 
the  total  stray  field^+A  [i  +-^f-}-  ^^1  +/2. 

L       -tViM      »i   j 

The  total  ampere-turns  of  the  field  magnet=the  difference  of 
magnetic  potential  between  the  pole  pieces + the  total  drop  of  potential 
in  the  field  magnet  corresponding  to  the  total  flux  =  the  fall  of  potential 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS       245 

in  the  armature  +  the  back  ampere-turns  of  the  armature  +  the  fall 
of  potential  in  the  air-gap -j- the  fall  of  potential  in  the  field  magnets 
corresponding  to  the  total  flux. 

When  the  alternator  is  unsaturated  or  but  slightly  saturated,  this 
latter  fall  of  potential  corresponding  to  the  total  flux  may  be  admitted 
proportional  to  the  flux 


Thus  the  flux  J/i  plays  a  part  entirely  similar  to  the  flux  /2,  and  it 
may  therefore  be  united  with  the  latter  in  the  coefficient  of  self- 
induction  of  the  armature.  It  must,  however,  be  remarked  that 
the  flux  J/!  only  follows  the  magnetic  circuit  to  the  point  of  emer- 

gence of  the  flux  from  the  field  magnets,  and  only  absorbs  consequently 
n 

the  fraction  —  —  of    the   ampere-turns   which   would   be   necessary 


to  make  it  traverse  the  entire  magnetic  circuit.  The  virtual  self- 
induction  5  which  may  be  advantageously  assumed  will  then  have 
an  approximate  expression 


\Rf2       Rtotal  RfJ ' 


and  this  should  be  employed  in  the  determination  of  the  segment 
BD,  as  above. 

In  order  to  get  the  total  necessary  field-magnet  ampere-turns 
.ZVi/i",  it  is  no  longer  necessary  to  add  any  excitation  upon  the 
field  magnets  except  the  ampere-turns  neutralizing  those  of  the 

armature    -  —  .     The   total   ampere-turns   XQ"   are    thus  ob- 

tained. If  the  armature  current  were  suddenly  suppressed,  an  E.M.F. 
E=Q"c  would  appear  in  the  armature  on  open  circuit,  which  is  that 
appearing  with  the  same  notation  in  Fig.  2. 

The  same  construction  may  serve  reciprocally  to  calcuate  the 
fall  of  potential  produced  in  an  alternator  having  the  excitation  XQ" 
for  the  reactive  or  wattless  current  Id  in  the  armature. 

Remark  No.  i,  Upon  the  Case  of  an  Unsaturated  Armature. 
When  the  armature  and  the  pole  pieces  are  not  saturated,  the  diagram 
of  E.M.F.  's  (Fig.  i)  is  also  a  diagram  of  the  flux,  to  a  different  scale, 
if  care  be  taken  to  divide  the  values  of  the  E.M.F.  's  by  the  coefficient 


\/2~ 


246 


METHODS  OF  CALCULATION 


Similarly,  having  given  the  magnetic  reluctances  sensibly  con- 
stant for  the  direct  flux  of  the  armature  and  for  the  transverse  flux 
(excluding  the  field  magnet  core),  the  same  diagram  may  also  present 
to  a  suitable  scale  the  M.M.F.'s  proportional  to  the  flux,  multiplied 
by  the  reluctance  of  the  armature,  of  the  air-gap,  and  of  the  pole 
pieces  respectively.  In  that  case,  OA  represents  the  necessary  ampere- 
turns  to  force  the  flux  through  the  said  reluctances.  OB  is  the  part 
of  this  M.M.F.  furnished  by  the  field  magnets;  AB  the  part  furnished 


D 


FIG.  5. 


FIG.  6. 


by  the  armature;   BD  the  supplement  necessitated  by  the  stray  fields 
of  the  armature.1     DF  represents  the  ampere-turns  of  distortion 


=  function  of 


V 


Calling  KI  a  coefficient  of  distortion,  or  of  transverse  reaction,  anal- 
ogous to  the  coefficient  K  of  the  direct  reaction,  and/,  (the  function) 

1  And  eventually  by  the  supplementary  stray  fields  Jft  in  the  particular  case 
indicated  above,  where  the  effect  of  the  stray  field  from  the  field  magnets  is 
referred  to  a  supplementary  term  of  the  armature  stray  fields. 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS        247 

the  relation  which  connects  the  ampere-turns  to  be  produced  by  the 
field-magnet  with  those  utilized  in  the  armature,  we  have  similarly 

FT=  function  of  ^^. 

\/   ry 

The  total  ampere-turns  necessary  to  the  emergence  of  the  flux  from 
the  field  magnets  will  be  determined  as  above  (Fig.  4):  let  XQ" 
represent  the  magnitude  equal  to  -ZVi/i",  to  the  scale  of  the  new 
figure;  the  distance  OQ  will  evidently  represent  the  ampere-turns 
absorbed  by  the  field  alone;  it  is  this  length  OQ  which  would  in 
general  be  corrected  by  taking  account  of  the  stray  field  Af\.  The 
method  of  correcting  the  stray  fields  indicated  above  may  be  em- 
ployed; we  will  give  further  on  another  more  nearly  accurate — already 
suggested,  moreover,  by  MM.  Picou  and  Guilbert.1 

Remark  No.  2,  upon  the  Subject  of  Diagram  No.  i.  It  is  to 
be  observed  that  following  the  respective  values  of  the  reactive  coeffi- 
cients, both  direct  and  transverse,  the  point  C,  the  extremity  of  the 
available  E.M.F.  on  open  circuit,  may  be  either  above  or  below  T. 

With  an  alternator  of  unsaturated  field  magnets,  the  two  reactions 
have  coefficients  nearly  equal,  and  C  may  then  coincide  with  71,  when 
they  are  equal.  Ordinarily,  the  coefficient  of  distortion,  Kt,  tends  to 
be  reduced,  as  we  shall  see,  by  a  reduction  in  the  breadth  of  the  poles, 
while  the  coefficient  K  has  the  opposite  tendency.  It  results  from 
this  that  C  tends  to  be  above  T.  But  the  saturation  of  the  field  magnets 
lowers  it  the  more  as  the  saturation  is  greater;  because  this  latter 
augments  but  slightly  the  supplementary  ampere-turns  necessary  to 
compensate  fof  those  of  the  armature,  but  greatly  diminishes  the 
variation  of  the  voltage  between  open  circuit  and  full  load.  C  is, 
therefore,  in  general  below  jT,  as  represented  in  the  diagrams. 

The  same  condition  may  be  found  even  with  saturation,  with 
certain  types  of  alternators,  such  for  example  as  that  which  was  exhib- 
ited in  1900,  in  Paris,  by  the  firm  Sautter-Harle.  This  machine, 
developed  along  a  plan,  formerly  patented  by  Professor  E.  Thomson, 
of  an  iron  rotor  (inductor  alternator),  has  a  single  armature,  two 
exciting  field-windings,  and  a  yoke  closing  the  magnetic  circuit  through 

1  The  diagram  of  the  present  figure  is  analogous  to  a  diagram  recently  pub- 
lished by  M.  E.  Guilbert  (loc.  cit.};  it  differs,  however,  in  that  the  line  FE  is 
expressed  as  a  function  of  the  coefficient  Kt  instead  of  the  coefficient  K,  and 
that  the  expression  of  dephasing  <J)  is  thus  presented  as  a  function  of  the  ampere- 
turns. 


248 


METHODS  OF  CALCULATION 


the  shaft  of  the  field  magnet.  It  presents  a  supplementary  air-gap  of 
considerable  reluctance  around  the  shaft,  and  this  air-gap  is  traversed 
only  by  the  direct  reaction.  The  coefficient  of  direct  reaction  is 
therefore  rendered  smaller  than  that  of  the  transverse  reaction,  and 
if  the  supplementary  stray  fields  J/i,  analyzed  above,  are  not  exag- 
gerated, C  will  remain  below  T. 

Those  theoretical  diagrams  are  not,  therefore,  liable  to  criticism 
which   show  C  below   T.     To  propose  placing  C  always  on   T,   as 


FIG.  7. 

Professor  Arnold  has  done,1  is  contrary  to  the  purpose  of  this  method, 
namely,  the  calculation  of  the  effects  of  saturation  and  of  distortion 
according  to  rational  principles. 

The  Case  of  a  Saturated  Armature.  (Fig.  7.)  The  theory 
of  two  reactions  permits  also  of  treating  the  case  of  a  saturated  arma- 
ture, by  employing  with  the  total  characteristic,  the  characteristic  of 
the  armature  alone  (comprising  the  armature,  air-gap,  and  pole  pieces), 

1  E.  Arnold,  Elek.  Zeit.,  1902,  page  250. 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS       249 

already  employed  moreover  by  MM.  Bauch,  Potier,  Guilbert,  and 
Picou. 

For  greater  clearness  of  explanation  I  shall  represent  the  char- 
acteristics upon  the  same  diagram,  but,  in  practice,  they  would  be 
drawn  upon  separate  figures.  The  curves  are  referred  to  the  E.M.F's. 
as  common  abscissae,  and  the  ordinates  of  the  two  curves  represent 
respectively  the  ampere-turns  for  the  passage  of  a  given  flux  (cor- 
responding to  the  E.M.F.)  in  the  magnetic  circuit,  with  or  without 
the  field  magnets.  The  difference  of  the  ordinates  equals  then  the 
ampere-turns  absorbed  by  the  field  magnet  alone,  in  the  absence  of 
stray  magnetic  fields. 

This  assumed,  we  shall  then  construct,  according  to  custom  ,the 
diagram  of  E.M.F. 's  by  adding  to  the  voltage  U  at  the  terminals  the 
internal  drop  r'l,  then  the  loss  by  stray  magnetic  fields  AF.  We 
thus  obtain  the  E.M.F.  OF  produced  in  the  armature.  We  find  upon 
the  characteristic  of  the  armature  the  ampere-turns  OF'  corresponding 
to  this  E.M.F.,  and  we  lay  them  off  on  OF'  in  the  direction  of  OF. 

We  may  then  observe  that  the  distortion  may  be  determined  by 
the  comparison  of  the  ampere-turns  of  distortion  with  the  useful 
ampere-turns  in  that  part  of  the  machine  which  does  not  include  the 
field-magnets;  because,  in  the  transverse  reaction,  the  reluctance  of 
the  pole  pieces  may  be  neglected  in  relation  to  that  of  the  entrefer 
and  of  the  armature  (especially  that  of  the  teeth),  and  consequently 
attribute  to  the  path  of  this  reaction  the  same  reluctance  as  in  the  path 
of  the  useful  flux  outside  the  field  magnets.  The  determination  of  the 
angle  <p  (Fig.  i)  will  then  be  transformed  into  simply  replacing  the 
self -inductance  by  the  M.M.F.'s  applied  to  the  armature.  The  ampere- 
turns  F'T=  — -T^r-  will  be  laid  off  in  the  direction  perpendicular  to 

V2 

7,  and  OT  is  joined;  then  from  F'  the  perpendicular  F'D  is  dropped, 
which  will  be  equal  to  the  ampere-turns  of  distortion 


The  line  OD  will  then  represent  the  ampere-turns  to  be  supplied  by 
the  field-magnets  at  the  point  of  emergence,  the  remainder  DF'  being 
furnished  by  the  armature  itself. 

It  remains  to  determine  the  total  field-magnet  ampere-turns  corre- 
sponding thereto,  taking  into  account  the  reluctance  of  the  stray  flux 


250  METHODS  OF  CALCULATION 

of  the  field  magnets,  properly  so  called.  For  this  purpose  it  suffices 
to  seek  upon  the  characteristic  of  the  armature  the  corresponding 
E.M.F.  db=e;  the  ordinate  Qb  corresponding  to  this  abscissa  e  upon 
the  curve  of  the  armature  measures  the  necessary  ampere-turns  to 
be  produced  between  the  poles,  the  necessary  stray-fields  /2  included. 
We  add  thereto  the  ampere-turns  equilibrating  the  direct  reaction  of 
the  armature,  that  is  to  say, 


V  2 

The  ordinate  Qq  represents  the  difference  of  potential  (subject  to  the 
factor  0.471)  necessary  between  the  poles  of  the  field  magnet.  From 
this  may  be  deduced  the  value  of  the  stray  field  /i  between  the  pole 
pieces,  which  may  be  presented,  for  example,  as  a  function  of  the 
magnetic  difference  of  potential  along  the  curve  XP,  which  is  sensibly 
a  straight  line;  Pp  will  then  represent  the  stray  field/!. 

If,  starting  from  b,  a  segment  bC  be  drawn  representing  Pp 
(measured  to  the  same  scale  as  the  flux  db  corresponding  to  the 
E.M.F.  e),  and  through  C  we  draw  the  straight  line  Cm  parallel  to 
OP,  the  latter  will  contain  between  the  two  characteristics  a  segment 
mn  which  will  represent  the  fall  of  magnetic  potential  in  the  field 
magnets  under  the  influence  of  the  total  flux  dC.  The  total 
necessary  M.M.F.  will  thus  be  equal  to  OP+mn. 

The  diagram  is  thus  established,  taking  into  account  the  stray 
field  both  of  the  field  magnet  and  of  the  armature.  It  is  distinguished 
from  those  of  Potier,1  Rothert,2  and  Bauch,3  because  it  takes  into 
account  the  transverse  reaction  with  its  real  value;  it  takes  account 
of  the  difference  between  the  two  coefficients  of  reaction  K  and  Kt 
and  is  thus  distinguished  from  the  diagram  of  M.  Guilbert4  for  unsat- 
urated  field  magnets;  it  finally  differs  in  diagrammatic  construction 
from  the  very  ingenious  diagram  of  the  same  author  for  saturated 
field  magnets  in  the  fact  that  it  does  not  separate  the  air-gap  from 
the  armature,  and  is  also  much  more  simple. 

Summing  up,  the  employment  of  the  diagram  in  Fig.  4  is  to  be 
recommended  for  the  case  in  which  there  is  no  appreciable  saturation 
either  in  the  armature  or  in  the  field.  In  all  other  cases  it  seems 

1  Potier,  Eclair  age  Electrique,  July  26,  1902. 

2  A.  Rothert,  Elek.  Zeit.,  1899. 
s  Bauch,  Elek.  Zeit.,  1902. 

4  M.  Guilbert,  Revue  Technique,  April  and  May,  1904. 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS       251 

preferable  to  employ  the  diagram  Fig.  7,  which  lends  itself  better  to 
determining  the  different  elements  without  complications. 

If  it  is  desired  to  solve  the  inverse  problem,  that  is,  to  determine 
the  fall  of  potential  under  constant  excitation  as  a  function  of  the 
load,  the  preceding  diagrams  do  not  give  a  direct  solution,  but  it  is 
easy  to  employ  them  for  an  indirect  solution,  particularly  in  assuming 
constant  the  external  dephasing  <p,  and  taking  successively  different 
values  of  the  reactive  current;  for  each  value  of  Id  the  preceding 
construction  will  be  followed  in  the  opposite  direction,  and  thus  will 
be  obtained  the  voltage  at  the  terminals,  the  values  of  (ft  and  of  Iw. 
Thus  may  be  traced  a  curve  of  voltage  u  as  a  function  of  Id,  and  of  / 
which  is  obtained  therefrom.  It  is  only  necessary  to  seek  upon  this 
curve  the  point  corresponding  to  the  conditions  required  and  the 
dephasing  angle. 

The  problem  is  solved  no  longer  for  a  single  point,  but  along  a 
complete  curve,  which  is  also  comparatively  easy. % 

Local  Corrections  of  the  Air-gap  Due  to  Saturation  (Second 
Approximation).  The  diagram  Fig.  7  is  established  by  supposing  that 
the  reactions  act  en  bloc,  and  are  represented  by  coefficients.  The  same 
is  true  of  the  diagram  Fig.  i .  But  if  it  be  desired  to  follow  the  reality 
somewhat  closer,  it  is  well,  once  the  diagram  is  determined  by  the 
aid  of  the  coefficient  K  (the  calculation  of  which  is  explained  later), 
to  calculate  upon  the  drawing  the  flux-density  of  the  resulting  field 
at  each  point  along  the  air-gap,  by  the  aid  of  magnetic  potential 
curves  (to  be  explained  later)  and  of  the  local  reluctance.  In  par- 
ticular, if  the  teeth  of  the  armature  are  saturated,  they  develop  marked 
variations  of  the  reluctance  per  unit  of  surface  along  the  air-gap,  and 
the  flux  calculated  according  to  a  mean  value  of  reluctance  may  be 
sensibly  modified  thereby.  This  is  the  case  not  only  for  the  trans- 
verse reaction,  as  has  already  been  remarked  by  certain  authors,  but 
also  for  the  direct  reaction,  which  should  not  be  set  aside  in  this 
correction.  The  effect  of  this  latter  is  to  reduce  the  resultant  flux. 
The  curve  of  the  diagram  is,  in  fact,  a  solution  of  the  first  approxi- 
mation necessary  in  order  to  determine  the  dephasing  of  the  values 
of  the  active  currents  from  the  reactive  currents.  After  these  values 
have  been  obtained,  a  second  approximation  may  be  arrived  at 
by  tracing  the  flux-densities  from  point  to  point  for  determining  the 
real  flux.  In  general,  however,  the  precision  of  the  calculations  is 
not  sufficiently  great  to  proceed  upon  this  correction  unless  ample 
time  may  be  afforded  for  the  study. 


252  METHODS  OF  CALCULATION 

Case  of  Field  Magnets  with  Divided  Windings.  In  certain  machines, 
particularly  turbo-alternators,  circular  field  magnets  are  found  in 
which  the  windings  are  carried  along  the  air-gap  in  slots  like  those 
of  the  armature.  The  preceding  diagrams  (Figs,  i  to  7)  apply  like- 
wise to  these  machines  only  on  the  condition  of  assuming  the  two 
coefficients  of  reaction  equal  even  when  the  field  magnet  is  entirely 
divided  into  slots.  Moreover,  the  field-magnet  winding  must  be 
affected  by  a  coefficient  of  ampere-turns  KI,  reducing  them  to  KiNiIi 
(with  Ki  =  o.4  to  0.5  in  completely  uniform  windings),  and  by  the 
Hopkinson  coefficient  v\t  which  is  calculated  like  the  stray  field  of  a 
slot  in  an  asynchonous  motor.  All  the  other  coefficients  are  calculated 
as  in  the  ordinary  case,  by  supposing  the  breadth  of  the  reactive  flux 
equal  to  that  of  the  field-magnet  poles.  This  method  has  given  me 
satisfactory  results  in  practice  for  this  type  of  machines. 

PART  II.     CALCULATION  OF  CONSTANTS. 

Practical  Calculation  of  Reactions.  In  order  to  apply  the  diagrams, 
the  coefficients  s,  /,  Kt  and  K  must  be  determined  (/  and  Kt  are  of 
course  only  two  expressions  of  one  and  the  same  coefficient).  The 
stray  coefficient  is  determined  by  known  methods  frequently  indicated 
for  asynchronous  motors,  and  they  need  not,  therefore,  be  alluded 
to  here.  For  K  and  Kt,  I  have  employed  for  several  years  the  most 
direct  method,  which  consists  in  determining  for  the  same  machine 
curves  of  distribution  of  magnetic  potential,  and  of  the  flux  in  the 
air-gap,  assuming  that  the  armature  is  traversed  by  a  known  current 
either  active  or  reactive.  By  taking  into  account  the  position  of 
the  pole  pieces  in  these  two  cases,  and  their  form,  as  well  as 
that  of  the  slots,  the  reactions  may  be  determined  with  sufficient 
precision. 

Let  us  consider,  for  example,  the  case  of  three-phase  currents: 
the  three  phases  occupy  in  a  double  field  six  slots,  or  groups  of  slots, 
and  at  the  passage  of  each  slot,  the  magnetic  potential  along  the 
entrefer  undergoes  a  sudden  positive  or  negative  increase  equal  to 

N- 
o.47T  multiplied  by  the  number  of  ampere -turns  —r-  contained  in  the 

slot.  It  suffices  to  mark  off  on  a  straight  line,  representing  the 
development  of  the  circumference  of  the  armature,  lengths  equal  to 
the  distance  between  the  axes  of  the  slots,  and  on  successive  ordinates, 
the  variations  of  the  magnetic  potential  thus  calculated.  The  horizontal 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS       253 

mean  line  of  the  curve  so  obtained  is  then  traced,  which  indicates 
the  zero  of  magnetic  potential.  The  fluxes  will  be  at  every  point 
proportional  to  the  ordinate  of  the  curve  from  the  zero  point,  and 
inversely  proportional  to  the  reluctance  per  unit  of  surface  correspond- 
ing to  the  abscissa  considered.  For  simplicity,  the  reluctance  may  first 
be  assumed  constant,  and  account  is  only  taken  of  its  variations  in 
order  to  arrive  at  a  definite  correction  for  a  second  approximation. 
To  simplify  the  calculation,  there  is  attributed  to  the  maximum  ampli- 
tude of  the  polyphase  currents  an  arbitrary  value  /o,  and  it  is  supposed 
that  the  currents  follow  a  sinsuoidal  law.  Since  the  form  of  the  curves 
is  reproduced  for  every  one-sixth  of  a  period  in  the  three-phase  cur- 
rents (or  for  one-fourth  of  a  period  in  two-phase  currents),  it  suffices 
to  study  them  during  such  an  interval,  and  then  to  outline  the  extreme 
forms. 


Let  us  take,  for  example,  a  three-phase  machine  with  six  slots  in 
the  field,  each  containing  N/6  wires,  calling  N  the  total  number  of 
peripheral  wires  per  double  field  (Fig.  8).  The  potential  produced 
by  each  is  0.27:  (N/6)i,  calling  i  the  current  which  traverses  the 
winding,  and  it  suffices  to  construct  the  curve  of  i,  to  which  that  of 
the  potentials  should  be  proportional.  We  take  two  positions;  one 


f  or  which  the  current  is  nil  in  the  slots  i  and  4  and  equals  ±  —  - 

in  the  others;  the  other  position  for  which  the  current  is  equal  to  IQ 
in  the  slots  i  and  4  and  equal  to  (1/2)  IQ  in  the  others.  The  curves 
proportional  to  the  potential  thus  obtained  are  respectively  represented 
in  Figs.  8  and  9.  On  these  figures  are  added  in  position  and  in 
magnitude  the  outlines  of  the  field-magnet  poles  in  the  two  positions; 
in  full  lines  the  position  corresponding  to  the  watt  current,  that  is  to 
say,  a  pole  axis  coinciding  with  the  middle  of  the  curve  of  potential; 


254  METHODS  OF  CALCULATION 

and  in  dotted  lines  the  position  corresponding  to  the  reactive  current, 
that  is  to  say,  the  axis  of  the  pole  facing  the  zero  of  the  curve. 

The  reaction  is  then  deduced  from  the  figure  by  determining  the 
mean  useful  ordinate  of  the  curve.  Theoretically  this  ordinate  would 
be  obtained  by  evaluating  the  area  of  the  shaded  curve  situated  in 
front  of  the  pole,  and  dividing  this  by  the  breadth  of  the  pole;  but 
the  result  so  obtained  is  not  practically  useful,  because  it  takes  no 
account  of  the  expansion  of  the  lines  of  force,  which  greatly  broadens 
the  flux,  particularly  as  the  air-gap  is  made  larger  and  the  angles 
of  the  pole  pieces  are  more  rounded.  To  determine  the  direct  reaction, 
one  must  take  instead  of  the  breadth  of  the  pole,  the  breadth  of  the 
field-magnet  flux  which  issues  from  it;  and  to  determine  the  mean 
ordinate  in  this  breadth.  A  similar  determination  is  made  for  the 
transverse  reaction.  It  must  be  observed  that  the  flux  which  forms 


FIG.  9. 

it  is  established  not  only  under  the  poles,  but  also  around  them, 
although  with  a  lesser  density.  Consequently,  this  flux  occupies 
a  greater  breadth  in  which  the  mean  reaction  should  be  determined. 
There  is,  therefore,  a  large  individual  liability  to  error  in  the  appre- 
ciation of  these  reactions,  and  this  should  give  preference  to  the  com- 
plete method  of  operation  here  indicated  for  the  employment  of 
theoretical  coefficients,  which  do  not  take  account  of  the  special  con- 
ditions in  each  machine.  If  the  breadth  of  the  flux  is  equal,  for  example, 

to  the  pitch,    Figs.    8    and    9    show  the    mean   ordinates  —  l/o— -— ) 

\J  *J 

and  §/o  for  the  direct  reaction,  and  similarly  for  the  transverse 
reaction.  The  values  give  those  of  the  coefficients  K  and  Kt  them- 
selves, if  the  ampere-turns  obtained  are  compared  with  the  ampere-turns 
which  would  be  obtained  with  the  three  bobbins  united  in  a  single 
pair  of  slots  and  traversed  by  a  current  70.  The  curve  of  potential 

2  N{   2     V/^\  N 

gives  -7~l'-7o — -)   instead  of  —  /0X3.     The  ratio  gives  the  coeffi- 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS       255 

Thus  the  coefficients  K  and  Kt  are  obtained  simply  by  taking  § 
of  the  mean  ordinates.  For  two-phase  currents,  one  would  similarly 
take  |,  that  is  to  say,  unity. 

If  instead  of  one  slot  per  phase,  there  were  several,  n  for  example, 
the  mean  ordinate  would  be  first  divided  by  n. 

In  this  manner,  the  following  figures  would  be  obtained: 


TABLE  IV 

EMPIRICAL  COEFFICIENTS 
Three-phase  winding,  with  three  separate  coils  per  pair  of  poles. 


Ratio  —  cf  the   breadth 

of  flux  to  that  of  polar 
pitch. 

COEFFICIENT  K  (DIRECT). 

COEFFICIENT  Kt  (TRANS- 
VERSE). 

Pos.  i. 

POS.    2. 

Mean. 

Pos.  i. 

Pos.  i. 

Mean. 

I         

0-385 
°-577 
0-577 

0.444 
0.500 

0-555 

0.419 
0.538 
0.566 

0.385 
0.288 
o.  192 

0.444 
0.333 
0.333 

0.419 
0.310 
0.2625 

2 

1     . 

In  practice  an  alternator  is  rarely  found  where  the  flux  occupies 
less  than  two-thirds  of  the  pitch,  and  besides  in  this  case  a  winding  of 
twelve  bobbins  with  six  short  slots  should  be  taken,  in  my  opinion, 
instead  of  the  ordinary  winding,  as  will  be  mentioned,  further  on. 

The  coefficients  of  self-induction  /  and  /',  corresponding  to  the 
two  reactions,  are  deduced  from  the  values  of  K  and  Kt  by  evaluating 
the  corresponding  fluxes  and  the  E.M.F.'s  which  they  induce  in  the 
windings  themselves  by  means  of  the  ordinary  formulas.  From  this, 
calling  k  the  winding  factor,  the  mean  value  of  /  which  takes  account 
of  the  reduction,  by  distribution  of  the  wires,  in  the  E.M.F.  pro- 
duced in  the  winding  by  a  sinusoidal  flux,1 


2 
—  I    10- 


1= 


•>(?) 


4xKk 


1  See  in  particular  the  coefficients  in   my   above-mentioned  analysis  of  the 
rotating  magnetic  fields,  Eclairage  Electrique,  1895. 


256  METHODS  OF  CALCULATION 

q  being  the  number  of  phases  (here  3),  Rt  and  Rd  the  reluctances  of 
the  transverse  and  direct  circuits  respectively.1 

These  reluctances  are  determined  from  the  drawing  of  the 
machine,  taking  into  account  the  real  path  of  the  lines  of  force  and 
the  saturation  of  the  parts  through  which  they  pass,  particularly  the 
teeth  of  the  armature,  the  polar  horns,  the  cores,  the  yokes  of  the 
field-magnets,  etc. 

If,  instead  of  alternate  poles,  the  machine  carries  poles  of  the 
same  name  (homopolar  inductor),  K  and  Kt  may  again  be  determined 
by  the  preceding  methods,  drawing  only  one  inductor-pole  for  two 
poles  of  the  armature.  It  results  from  this  that  theoretically  the 
reactions  would  give  rise  to  coefficients  50  per  cent  less  than  in  the 
ordinary  case.  In  practice,  however,  this  is  far  from  being  the  case, 
because  of  the  very  considerable  expansion  of  the  flux  of  the  arma- 
ture in  the  large  spaces  existing  between  the  field-magnet  poles. 
The  direct  flux  reaction  and  particularly  the  transverse  reaction  is, 
therefore,  much  larger  than  if  they  were  produced  only  by  the  action 
of  the  poles;  so  that  finally  the  reactions  are  scarcely  reduced  more 
than  25  per  cent.  The  stray  fields  are,  moreover,  very  large  in  this 
type  of  machine,  and  every  expansion  of  the  field-magnet  flux  beyond 
the  breadth  of  the  pitch  produces  a  hurtful  inverse  E.M.F.  The 
induction-density  in  the  fcir-gap  should  finally  be  doubled  at  least, 
to  produce  the  same  useful  flux  on  open  circuit.  From  all  the  above 
it  follows  that  homopolar  machines  are  of  little  advantage  and  are 
almost  abandoned. 

To  completely  take  into  account  the  practical  values  of  the 
coefficients,  we  shall  consider  again  the  case  of  three-phase  machines 
with  six  coils  per  field,  first  concentrated  into  one  pair  of  slots  per 
coil,  and  then  spread  uniformly  (or  to  a  large  number  of  slots  each) 
in  order  to  occupy  the  entire  circumference  of  the  armature. 

Figs,  ii  and  12  represent  the  curves  of  magnetic  potential 
obtained  in  the  two  hypothetical  cases  with  long  bobbins  disposed 
as  shown  diagrammatically  in  the  figure.  The  two  curves  corre- 
spond to  the  same  hypothesis  as  above  for  currents,  and  Table  II 
represents  the  separate  mean  values  obtained  for  K  and  Kt  from 
these  curves. 

1  In  unsaturated  alternators,  if  one  calls  e  the  single  air-gap,  s  the  polar  surface, 
and  d  the  coefficient  of  enlargement  of  the  flux,  the  equation  is  approximately 

.    i      s(i  +  d) 
obtained  =-= '. 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS         257 


Winding 
Phases 


-2-2      1      1  —  3-3      2      2-1-1     8      8-2-2      1      1-3—82      2 
FIG    10 


Fos.l 


Pos.  2 


11     l — n> 

-+-H--4-U 


"Windine 
*ma    lg 


___ 
3  —1      2-81   -2'     3'-l      2  -3'    l'  -2      8  -l'     2'-3      1  -2'    8'  -1      2'-s' 

FIG  13 


FIGURES  10  TO  15 


258 


METHODS  OF  CALCULATION 


But  a  winding  with  six  coils  may  also  be  realized  symmetrically 
following  the  plan  of  Fig.  16,  from  which  two  new  curves  i6a  and  i6b 
are  obtained.  The  table  also  indicates  the  value  of  the  coefficients 
thereby  deduced. 


TABLE  V 

EMPIRICAL  COEFFICIENTS 

Three-phase  winding  with  three  separate  coils  per  pair  of  poles. 


Rat 

.      8 

K. 

Kt. 

10  J 

Pos.  i. 

POS.    2. 

Mean. 

Pos.  i. 

POS.    2. 

Mean. 

o   381; 

o  380 

0     ^87 

o   38^ 

O     T.&O 

8 

Long  coils  • 

§  

0-505 

o-577 
o.c886 

0.500 

0-555 
o  2777 

0.5025 
0.566 
o  283 

0.288 

o.  192 
o  2886 

0.291 
0.222 
O   2777 

0.2915 
0.207 
o  28^ 

4 

o   360 

o  37  ^ 

o   367  ^ 

o  °  16 

o  208 

*  

0-385 

0.444 

0.415 

o.  192 

0.  Ill 

o.  152 

TABLE  VI 

EMPIRICAL  COEFFICIENTS 
Three-phase  distributed  winding,  3  or  6  long  coils  per  pair  of  poles. 


Ratio  -?-  breadth  of 

K. 

Kt. 

J 

flux  to  polar  pitch. 

Pos.  i. 

POS.    2. 

Mean. 

Pos.  i. 

POS.    2. 

Mean. 

i  

0.385 

0.389 

0.387 

0.385 

0.389 

0.387 

f  
1 

0.505 

O     ^3 

0.500 

O    ^4.1 

0.5025 
O    ^4.7 

0.2886 

O    2  l6 

0.2916 
O    2  36 

o  .200 

O    2  36 

Here  again  it  is  seen  that  the  direct  reactions  increase  while  the 
transverse  reactions  diminish  when  the  breadth  of  the  flux  (larger 
than  the  pole)  diminishes.  It  is  moreover  determined  that  the  reac- 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS        259 


TABLE  VII 

EMPIRICAL  COEFFICIENTS 

Three-phase  winding  with  6  short  distributed  coils  per  double  field. 


Ratio    9 

K. 

Kt. 

Pos.  i. 

POS.    2. 

Mean. 

Pos.  i. 

POS.    2. 

Mean. 

1  
-1 

0.289 
o   360 

0.277 

O    37^ 

0.283 
o    367  ^ 

0.289 
O    2  l6 

0.277 

o  208 

0.283 

O    2  12 

J  

0.385 

0.444 

0.4145 

o  .  192 

0-137 

o  .  1645 

tions  are  markedly  reduced  by  the  employment  of  short  coils.  But 
it  may  be  readily  shown  that  with  a  sinsuoidal  field-magnet  flux  of  a 
breadth  equal  to  the  pitch,  the  E.M.F.  induced  in  the  winding  is 
reduced  approximately  in  the  same  ratio.  In  fact,  the  mean  breadth 
of  the  short  bobbins,  Fig.  16,  is  J  of  the  pitch,  while  the  mean  breadth 


FIG.  16. 

of  the  long  bobbins,  Fig.  10,  is  J  the  pitch.     The  straddling  arrange- 
ment of  coils  in  Fig.   10  involves  only  the  coefficient  of  reduction 

71 
2  COS 


=  0.966,  while    the    arrangement    of  Fig.   16    reduces  the 

z 

E.M.F.  in  the    same  ratio  as    the  flux   linkage,  that  is    to  say,  by 


71        \/2 


the  coefficient  &=sin  — = 

4        2 

E  short  bobbins    0.707 

=  — ^—=0. 

E  long  bobbins     0.966 


The  two  E.M.F.'s  are  then  in  the  ratio 


260 


METHODS  OF  CALCULATION 


But  the  ratio  of  the  coefficients  K  and  Kt  of  the  two  windings  gives 
approximately  the  same  figure.     It  is  to  be  observed  in  this  connection 


FIG.  i6fl. 


FIG.  i6b. 


FIG.  i6c. 


that  for  the  case  of  a  flux  having  a  breadth  equal  to  the  polar  pitch, 
the  coefficient  K  is  substantially  equal  for  these  two  windings,  that 

is,  to  k  of  each  winding  multiplied  by  ( —  j  ,  which  verifies  the  general 


THE  ARMATURE  REACTIONS   OF  ALTERNATORS        261 

law  formerly  announced  and  which  is  alluded  to  above.1  Hence  it 
follows  that  the  fluxes  produced  by  multiple  -coil  windings  differ  but 
little  from  the  mean  value  of  the  theoretical  fluxes,  and  approach  the 
more  nearly  as  the  sections  are  the  more  numerous,  and  also  as  the 
local  variations  or  fluctuations  of  the  curves  between  the  extreme 
forms  i  and  2  of  the  appended  figures  are  damped  out  by  the  Foucault 
currents  of  the  neighboring  pole  pieces.  The  energy  expended  in 
these  Foucault  currents,  being  supplied  by  the  armature,  is  represented 
by  an  augmentation  of  its  apparent  resistance  r'. 

The  reactions  of  two-phase  armatures  would  be  found  in  a  similar 
manner,  and  it  will  not  be  necessary  to  reproduce  them  more  in  detail. 
Moreover,  two-phase  machines  are  more  and  more  becoming  sup- 
planted by  three-phase,  and  the  latter  present  reactions  of  much 
smaller  fluctuations  and  a  better  utilization  of  materials,  just  as  three- 
phase  motors  are  superior,  from  this  standpoint,  to  two-phase  motors. 

Comparison  with  Theoretical  Coefficients.  The  theoretical  coeffi- 
cients are  easy  to  establish  in  the  case  where  a  sinusoidal  flux  is 
assumed  and  the  harmonics  are  suppressed.2  It  is  then  demonstrated 
that  the  magnetic  potential  produced  by  a  polyphase  winding  of  q 
phases  is  independent  of  the  number  of  phases  and  depends  only  upon 
the  total  number  of  wires  N  per  double  field,  and  that  it  is  repre- 
sented by  a  sinusoid  whose  amplitude  is  2NIQ.  The  mean  potential 

2 

in  the  air-gap   is  therefore  —  zNIs  and  the  equivalent  mean  magneto- 

7T 

motive  force  producing  the  reactive  flux  on  closed  circuit 


that  is  to  say,  (4/7r)2  of  the  magnetomotive  force  which  will  give  the 
same  turns  if  they  coincide  in  position  and  phase.  In  this  case  the 
sinusoid  of  potential  (Fig.  17)  is  entirely  used  and  the  direct  reaction 
is  proportional  to  the  mean  ordinate  of  the  area  A^B,  and  the  trans- 
verse reaction  to  that  of  the  area  $HBH'io.  If,  on  the  contrary, 

1  In  alternators  with  distributed  windings  (Figs.   12   and   15),   the  diagrams 
of  windings  10  and  13  respectively  may  be  employed  by  assuming  that  each  coil 
is  replaced  by  a  zone  of  wires  occupying  along  the  air-gap  a  breadth  of   iV  of 
the  field,  of  £  of  the  polar  pitch,  and  having  as  median  line  the  old  outline  of 
the  single  bobbin  which  it  replaces. 

2  See  my  above-mentioned  memoir  of  1899  upon  "Rotating  Magnetic  Fields," 
see  also  Arnold  and  la  Cour's   "  Vorausberechnung  der  Ein  und  Mehrphasen- 
stromgeneratoren."     Stuttgart,  1901. 


262 


METHODS  OF  CALCULATION 


the  reactive  flux  only  occupies  a  part  d  of  the  pitch  instead  of  J,  as  indi- 
cated by  the  dotted  intersecting  lines  in  the  figure,  the  reactions  will  be 
proportional  to  the  mean  ordinates  of  the  areas  12345  and  67.689 
respectively,  limited  to  the  breadth  of  the  flux  (which  may  be  different 
moreover  for  the  transverse  flux  from  what  it  is  for  the  direct  flux. 
By  integrating  the  area  of  the  sinusoid  from  H$  on  to  54  one  finds 

Ordinate  of  area  12345  =  —  sin  (  — - }  X—  /o- 

0  \2    J/         7T 

Ordinate  of  area  6fjBSg=  -I  i— cost  — -rj    X—  /o- 


FIG.  17. 

The  coefficients  which  apply  to  the  ampere-turns,  resulting  from 
a  restriction  of  the  flux,  are  then  respectively  for  K  and  Kt 


d 
for-r=j; 


^^       and     C<=I~COV 
and  have  the  following  values  for  example  (not  including  k) 

/2\ 
1.232;     ^  =  0.817  from  which  K^=i.2^2\  —  } 

\7T/ 


=§;    c=  1.299;    ^=0.75 


c=  1.414;    ^=0.586 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS        263 

TABLE  VIII 

THEORETICAL  COEFFICIENTS 
Winding  with  three  coils  per  pair  of  poles. 


Ratio  — 
J 

SEPARATE  COILS. 

DISTRIBUTED  COILS. 

K. 

Kt. 

K. 

Kt. 

i 

0.405 
0.526 
0.572 

0.405 
0.304 
0.237 

0-3859 
0.501 

0-544 

0-3859 
0.2897 
0.2257 

1 

1 

TABLE  IX 

THEORETICAL  COEFFICIENTS 
Winding  with  6  coils. 


Ratio  j 

SEPARATE  COILS. 

DISTRIBUTED  COILS. 

K. 

Kt. 

K: 

Kt. 

r*  

0.391 

0.391 

0-3859 

0-3859 

Long  coils.  .  .  • 

0.508 

0.293 

0.501 

0.2897 

1 

0-552 

0.228 

0-544 

0.2257 

rl 

0.286 

0.286 

0.2825 

0.2825 

Short  coifs  .  .  .  • 

0.371 

0.215 

0.3665 

0.212 

0.404 

o.  167 

0.399 

o.  164 

Since  these  figures  must  also  be  modified  in  every  case  by  the  coeffi- 
cient k  corresponding  to  the  E.M.F.,  as  has  been  explained  above, 
it  is  evident  that  they  will  differ  but  little  from  those  of  Table  II. 
But  it  is  not  the  less  necessary  to  determine,  from  a  drawing  of  the 
machine,  the  breadth  of  the  flux  before  applying  it  in  these  formulas, 
and  therefore  to  correct  the  reactions  in  order  to  take  account  of  the 
saturation  of  the  different  parts  of  the  armature  and  of  the  pole  pieces. 
This  correction  is  made  by  assuming  the  magnetic  conditions 
which  are  approximately  attained  in  the  machine  at  full  load.  The 


264  METHODS   OF  CALCULATION 

mean  flux-density  is  then  known  which  must  be  developed  in  the 
air-gap,  the  teeth,  and  the  pole  pieces,  and,  moreover,  one  has  from 
the  curves  of  potential  the  value  of  the  magnetomotive  forces  acting 
at  all  points  of  the  air-gap.  From  this  may  be  deduced  the  real 
flux-density  at  every  point,  and  consequently  the  true  variation  of 
the  total  flux  produced  by  the  reaction.  These  expressions  of  self- 
inductions  given  above  should  be  consequently  replaced  by  the  integrals 
of  the  form 

N 


indicating  by  x  the  abscissa,  and  by  y  the  ordinate  of  the  curve,  and 
R  the  reluctance  per  unit  of  surface  at  this  point,  b  represents  the 
length  of  the  armature,  q  the  number  of  phases,  N  the  number  of 
peripheral  wires  per  field. 

Case  of  Single-phase  Alternators.  The  problem  of  the  reactions 
of  single-phase  alternators  is  more  complex  than  that  of  polyphase 
alternators,  as  will  be  seen.  It  does  not  appear  to  have  been  fully 
understood  by  the  authors  who  have  previously  treated  it.  It  may 
be  analyzed  by  the  same  method  as  that  which  I  have  formerly  developed 
for  asynchronous  motors,1  but  taking  into  account  this  important 
difference,  that,  in  general,  the  rotating  reactions  (rotating  magnetic 
field  with  respect  to  the  armature,  which  we  suppose  fixed),  are  sup- 
pressed in  motors  by  the  short-circuited  windings  on  the  rotor,  but 
not  in  alternators,  except  in  the  case  where  they  are  furnished  with 
massive  poles,  or  especially  with  the  damping  windings  of  Leblanc 
(which,  however,  only  give  a  partial  suppression). 

I  have  shown  that  each  coil  of  a  single-phase  armature  produces 
a  reactive  flux  capable  of  being  decomposed  in  space  into  sinusoidal 
harmonics,  of  which  the  first,  the  only  one  which  need  be  considered 

in  practice,  has  for  amplitude  -^ — ,  denoting  by  nl,  the  ampere- 
turns  of  the  coil,  and  by  R,  the  reluctance  of  the  magnetic  circuit 
traversed  by  the  flux  which  it  produces.  In  accordance  with  the 
theorem  of  Leblanc,  I  decompose  this  pulsating  sinusoid  into  two 
rotating  sinusoids  of  one-half  amplitude:  one  turning  synchronously 
with  the  rotor,  and  not  displaced  therefore  with  respect  to  the 
field -poles;  it.  provides  only  a  fixed  reaction;  the  other  rotate 

1  Blondel,  "Properties  of  Rotating  Magnetic  Fields,"  Eclair  age  ElectHque, 
May,  1908. 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS        265 

in  the  opposite  direction  with  the  same  speed  and  is  therefore 
displaced  with  respect  to  the  field  magnets  with  a  speed  double  that 
of  synchronism;  this  gives  rise  to  a  pulsating  flux  in  the  field  magnets 
of  frequency  double  that  of  the  e.m.f.  induced  in  the  armature,  as 
I  have  also  shown  experimentally.1  This  reaction  is  somewhat  weakened 
by  the  currents  which  it  produces  in  the  closed  circuit  of  excitation, 
but  it  is  not  completely  extinguished  and  produces  therefore  an  e.m.f. 
of  normal  frequency  in  the  armature  which  should  be  taken  into  con- 
sideration. But,  while  the  fixed  reaction  may  be  analyzed,  exactly 
as  in  polyphase  alternators,  into  direct  and  transverse  reactions,  the 
parasitical  rotating  reaction  is  effected  directly  through  the  field  mag- 
nets, as  well  as  transversely  through  the  pole  pieces,  and  may  be 
represented,  consequently,  by  a  mean  coefficient  of  self-induction 
similar  to  the  self-induction  of  the  stray  field  ais,  to  which  it  is  added. 
It  may  be  directly  demonstrated  by  calculations  that  this  analysis 
readily  lends  itself  to  the  interpretation  of  the  facts,2  and  that  a 

single-phase  alternator  whose  armature  at  rest  presents  a  self-induction 

/ 

1  Blondel,   "Photographic  Record  of  Periodic  Curves,"   Lumiere  Electrique, 

August,  1891. 

2  In  fact,   the  self-induction  of  the  armature  L,   varying  between  the  two 
values  A  and  X'  according  to  position,  may  be  represented  by  an  expression 

L=A+B  coszcot  +  cos 
and  abbreviating,  A+B=Xf;  A-B=X, 

indicating  always  by  s  the  inductance  of  the  stray  fields  in  the  slots.  Let 
e=E0sintot  be  the  internal  E.M.F.,  and  i=I0sm(cot  —  ^)  the  strength  of  the 
current,  and  there  is  immediately  obtained  as  the  difference  of  potential  at  the 
terminals  of  the  machine  the  following  expression: 

rdi     .dX         di 
u-e-H-L#-*dt-«*tf 

Replacing  e,  L,  i  by  their  values,  and  neglecting  a  term  — cos  ($(ot—  </>)  which 
produces  an  upper  harmonic,  there  remains 

(R\ 
A j  70  cos  ^  cos  tot 

—  oj  (A  -\ — J  70sin  ¥  sin  cot  —  cosl^  cos  (cut  —  ty) 
=  E0  sin  cot  —  rIQ  sin  (cot  —  ty) — -70  cos  ( cot — SI>)  —  — 70  cos  ty  cos  cot 

70  sin  ty  sin  cot  —cJsI0  cos  (co  —  &). 

%  A' 

The  values  of  —    and  —  representing   that  which   we  have   uniformly  denoted 

by  /  and  /'  for  all  the  machines  in  the  construction  of  the  diagrams. 


266  METHODS  OF  'CALCULATION 

^  when  the  poles  are  crossed,  and  an  inductance  of  X'  when  they  are 
coincident,  behaves  under  load,  when  the  real  dephasing  is  0  for 
example,  as  though  the  active  current  /  cos  ^  traversed  an  inductance 

—  ,  the  reactive  current  7  sin  d>  an  inductance  —  ,  and  the   total  cur- 


rent  I  a  parasitic  inductance  -    —  equal  to  the  mean  of  the  two 

preceding^ 

In  order  to  take  into  account  the  Foucault  currents  of  the  arma- 
ture produced  by  the  rotating  reaction,  it  is  sufficient  to  apply  a 
reducing  coefficient  m  to  its  inductance,  which  is  less  than  unity, 
evaluated  according  to  the  conditions  of  construction,  and  at  the  same 
time  to  increase  the  apparent  resistance  of  the  armature  in  accordance 
with  the  energy  lost  in  these  Foucault  currents,  since  it  is  furnished 
by  the  armature.  It  is  for  this  reason  that  we  attribute  to  this 
resistance  a  value  r  '  >  r  in  all  of  our  diagrams. 

If  the  field  magnets  or  the  armature  of  the  alternator  are  saturated, 
the  direct  and  transverse  self  -inductances  will  be  replaced  by  equiva- 
lent back  ampere-turns,  again  calculated  as  in  my  theory  of  rotating 
field:  the  sinusoid  of  the  amplitude  o.2nnl  presents  the  mean  ordinate 

7  /     \  2 

-  times   smaller  and  is  consequently  equivalent  to  —  I  —  j    ampere- 

/2\2 

turns.     The  value  of  K  and  Kt  will  therefore  be  (  —  J    for    a    single 

bobbin  if  it  has  the  same  breadth  as  the  flux.  If  the  winding  com- 
prises several  straddling  coils,  the  coefficients  should  be  multiplied 
by  the  straddling  factor  of  the  winding  k\  finally  if  the  flux  of  the 
poles  is  narrower  than  the  pitch,  it  should  be  multiplied  by  the  factors 

—  sin  (  —  T  )  and  -r-  i  —  cosf  —  7)  ,  respectively  the  coefficient  of 
d  \2  J/  ^L  \2  A)\ 

direct  reaction  and  the  coefficient  of  distortion  for  the  conditions 
analyzed  above.1 

1  It  is  of  course  easy  to  pass  from  a  single-phase  machine  to  a  polyphase 
machine  of  two  phases,  observing  that  each  phase  gives  a  fixed  reaction  and  a 
rotating  reaction  of  the  same  amplitude.  I  have  shown  in  my  theory  of  rotating 
fields,  already  alluded  to,  that  the  fixed  reactions  unite  in  space  and  are  added 
algebraically  while  the  q  rotating,  parasitic  reactions  give  rise  to  a  resultant  zero. 
The  coefficients  K  and  Kt  are  then  themselves  theoretically  expressions  in  all 
the  machines  independent  of  the  number  of  phases  (AT"  designating  always  the 
total  number  of  peripheral  wires);  but  the  rotating,  parasitic  self  -inductance 
disappears  in  polyphase  machines.  In  this  manner  the  return  is  made  to  the 
theoretical  coefficients  of  the  polyphase  machines. 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS       267 

To  sum  up,  the  case  of  the  single-phase  alternator  should  then 
be  treated  exactly  by  the  same  general  formulae,  the  same  construc- 
tions, and  the  same  diagrams  as  in  the  case  of  a  polyphase  alternator, 
but  under  the  condition  of  considerably  increasing  the  stray  flux 
ajsl,  adding  thereto  a  term  representing  the  parasitic  rotating  in- 
ductance. The  coefficient  cos  is  thus  replaced  by  w  s  +  w(  —  -J 

values  /  and  V  are  calculated  by  the  theoretical  coefficient  of  poly- 
phase machines  (see  p.  263)  (taking  into  account  the  saturation  of 
the  circuits  by  the  values  given  to  R  and  Rf,  as  has  already  been 
seen  above).  In  this  manner  a  coefficient  m  of  reduction  will  be 
determined  less  than  unity,  the  more  or  less  marked  suppression  of 
the  parasitic  rotating  inductance  by  the  Foucault  currents  induced 
in  the  surrounding  non-laminated  metallic  masses,  and  in  some  cases 
in  special  damping  circuits. 

Consequences  from  the  Point  of  View  of  the  Construction  of 
Alternators  for  Good  Regulation.  The  theory  and  the  calculation 
for  the  reactances  just  as  they  have  been  above  analyzed,  lend  them- 
selves to  the  discussion  of  the  construction  of  alternators  much  better 
than  the  old  methods.  We  will  proceed  to  give  a  few  examples  of 
such  applications. 

(A)  In  so  far  as  concerns  the  employment  of  short  bobbins,  (gener- 
ally abandoned  by  reason  of  the  disfavor  thrown  upon  them  by 
windings  of  only  three  coils  per  field,  which  give  terrible  pulsations), 
the  winding  of  six  short  bobbins  which  I  have  indicated  (Fig.  13) 
may  be  compared  with  the  ordinary  winding  (Fig.  10)  by  means  of 
the  coefficients  calculated  above.  The  relations  between  the  K  of 
the  two  being  the  same  as  between  the  respective  coefficients  k  of  the 
E.M.F.,  it  is  evident  that  the  advantage  of  the  short  bobbins  from 
the  point  of  view  of  reactions,  involves  a  loss  of  E.M.F.  in  exactly  the 
same  proportion.  To  re-establish  the  desired  value  of  the  latter,  the 
number  of  turns  of  the  armature  must  either  be  increased,  consequently 
re-establishing  the  same  reaction,  or  the  flux  density  in  the  air-gap 
must  be  increased,  and  consequently  the  ampere-turns  of  the  field- 
magnets  as  well  as  the  losses  by  Foucault  currents.  The  two  windings 
are  therefore  equivalent  from  the  constructive  standpoint  when  the 
field -magnet  flux  occupies  the  entire  polar  pitch.  However,  short 
bobbins  may  be  treated  more  rationally  by  reducing  the  breadth  of 
the  field-magnet  flux  to  two-thirds  of  the  field-magnet  pitch  in  such  a 
manner  that  the  flux  shall  be  entirely  utilized  in  the  coil.  Further 


268  METHODS  OF  CALCULATION 

increasing  the  flux-density,  an  e.m.f.  is  obtained  equal  to  that  in 
long  bobbins,  and  the  direct  reaction  remains  in  the  same  ratio  with 
respect  to  the  e.m.f.  while  the  transverse  reaction  is  reduced.  The 
winding  of  Fig.  13,  with  a  flux  having  the  value  of  two-thirds  of  the 
pitch,  is  then  that  which  for  a  given  ratio  of  counter-ampere-turns 
of  the  armature  to  the  ampere-turns  of  excitation,  produces  the  smallest 
transverse  reaction.  This  reduction  is  of  considerable  importance. 

(B)  As  to  the  methods  of  reducing  reactions,  it  results  from  the 

•ty 
preceding   that  there   is  no   means  of   reducing   the   ratio   — .     The 

rv 

only  means  of  improving  the  regulation  of  alternators  are  therefore, 
first,  to  saturate  the  field-magnet  circuit  (which  augments  the  m.m.f. 
and  reduces  the  variations  of  e.m.f.  at  the  armature  terminals  as  a 
function  of  the  reactive  current);  second,  to  increase  the  air-gap, 
which  is  less  effective;  and  third,  to  reduce  the  transverse  reaction, 
which  has  the  effect  of  diminishing  the  dephasing  of  the  diagrams, 
and  consequently  the  direct  reaction,  which  is  proportional  to  the  reac- 
tive current.  An  alternator  which  would  not  have  transverse  reaction, 
would  have  nothing  to  fear  from  direct  reaction,  even  if  it  were 
enormous. 

The  methods  of  reducing  the  transverse  reactions  alone  are:  First, 
the  reduction  of  the  breadth  of  the  flux  accompanied  by  an  augmen- 
tation of  the  flux-density  in  the  air-gap,  re-establishing  the  same 
field-magnet  flux  at  the  expense  of  the  increase  of  the  flux-density; 
but  in  that  case  the  same  result  would  be  obtained  by  a  simple 
increase  of  the  air-gap,  without  an  increase  in  the  loss  of  energy 
in  the  teeth;  second,  the  saturation  of  the  polar  horns  when  the  pole 
pieces  possess  them;  third,  the  addition  of  longitudinal  slots  in  the 
field-magnets,  as  in  direct-current  dynamos.  This  last  method  is 
very  effective  when  the  field  magnets  are  saturated  and  the  slots 
occupy  their  entire  length  and  are  continued  partially  into  the  yoke; 
the  reduction  of  distortion  thus  obtained  involves  generally  a  reduction 
of  the  total  flux,  because  the  mean  permeability  of  the  field  magnet 
is  reduced  by  the  inequality  of  the  m.m.f. 's  established  between  its 
two  halves;  but  the  augmentation  of  excitation  which  results  is  neg- 
ligible in  comparison  with  the  diminution  obtained  in  the  reactive 
current  by  the  reduction  of  dephasing. 

(C)  As  to  the  comparison  between  single-phase  and  polyphase 
alternators,  it  is  seen  that  not  only  do  single-phase  alternators  utilize 
less  effectively  their  materials  for  the  production  of  energy,  because 


THE  ARMATURE  REACTIONS  OF  ALTERNATORS       269 

their  armature  surface  is  less  utilized  for  e.m.f.,  but  also  their  arma- 
ture reaction  gives  rise  to  a  hurtful,  parasitic  self-induction  which 
does  not  exist  in  polyphase  alternators  and  which  reduces  their  good 
regulation.  This  parasitic  inductance  can  only  be  partially  suppressed 
at  a  cost  of  the  expenditure  of  energy  equivalent  to  a  considerable 
augmentation  of  the  apparent  resistance  of  the  armature. 

RESUME  AND  CONCLUSION 

To  sum  up,  it  has  been  established  in  this  paper  that  the  theory 
of  two  reactions  of  the  armature  admits  of  analyzing  the  phenomena 
of  alternators  with  greater  precision  than  the  old  theories,  besides 
having  the  advantage  of  referring  them  to  conditions  similar  to  those 
of  direct-current  machines.  Simple  diagrams  are  given  applicable  to 
alternators  of  saturated  field  magnets  and  unsaturated  armatures 
(Fig.  4)  and  even  to  saturated  armatures  (Fig.  7)  without  involving 
a  complicated  correction. 

It  has  been  indicated  how  to  calculate  the  coefficients  of  reaction, 
not  only  theoretical,  but  also  actual  values,  by  means  of  curves  of 
magnetic  potential  in  the  air-gap.  Interesting  relations  have  been 
established  between  these  coefficients  and  those  of  the  induced  e.m.f. 

Comparisons  have  been  established  between  the  different  types  of 
winding,  and  the  advantages  possible  for  a  special  winding  with  short 
bobbins  have  been  made  evident. 

It  has  been  shown  that  single-phase  alternators  may  be  treated 
by  the  same  methods,  adding,  however,  to  the  inductance  of  the  stray 
field  a  parasitic  inductance  which  does  not  exist  in  polyphase  machines. 

Finally  the  consideration  of  the  transverse  reaction  has  permitted 
the  discussion  of  a  construction  in  view  of  good  regulation,  showing 
the  interest  which  attaches  to  reducing  the  coefficient  of  distortion 
in  alternators,  and  indicating  the  means  of  such  'reduction. 

The  author  hopes  that,  thanks  to  simplicity  of  application,  much 
greater  than  is  often  believed,  and  by  its  relations  with  the  theory 
of  direct-current  machines,  this  method  of  calculation  (in  which  he 
has  had  practical  experience  for  several  years)  may  be  of  service  to 
designers  and  satisfy  the  need  of  rational  precision  in  this  work. 


CHAPTER  II 

METHODS    OF    TESTING    ALTERNATORS    ACCORDING    TO    THE 
THEORY  OF  TWO  REACTIONS  * 

f  i 

BY  PROF.  ANDRE  BLONDEL,  Ecole  des  Fonts  et  Chaussees. 

THE  author  described  in  the  Bulletin  de  la  Societe  des  Eledriciens 
in  1892  a  method  of  testing  alternators  similar  to  that  of  Hopkinson 
for  direct-current  machines,  which  permits  of  studying  both  their 
efficiency  and  their  armature  reactions,  under  the  same  conditions 
of  operation  and  without  a  large  expenditure  of  power,  under  the 
sole  condition  that  the  two  alternators  shall  be  similar.  This  method 
depends  upon  the  rigid  coupling  of  two  similar  alternators.  Later 
the  author  published  a  variation  of  the  method  which  does  not  call 
for  the  rigid  coupling  between  the  alternators,  but  which  consists 
in  operating  the  alternator  on  test  as  a  synchronous  motor,  by  the 
aid  of  an  auxiliary  alternator.2  The  alternator  under  test  revolves 
on  no-load,  as  a  motor,  under  the  normal  current.  This  method  has 
been  designed  particularly  to  determine  the  efficiency  with  greater 
precision  than  by  separating  the  various  losses.  The  object  of  the 
present  paper  is  to  complete  and  yet  further  perfect  this  method  by 
pointing  out  how  it  may  be  likewise  used  for  measuring  armature- 
reactions  (especially  by  the  employment  of  two  reactions,  set  forth 
in  the  preceding  chapter). 

Method  No.  i.  When  the  Rigid  Coupling  of  the  two  Alternators 
'is  Possible.  When  two  similar  alternators  are  available,  and  when 
they  can  be  placed  side  by  side  so  as  to  be  connected  rigidly  by  a 
short  coupling,  the  following  tests  may  be  carried  out  (Fig.  i): 

First  a  certain  difference  of  phase  a  is  provided  between  the 
alternators  (for  example,  a  phase-difference  of  30°,  that  is  to  say, 
one-sixth  of  a  pole,  or  45°,  that  is  to  say,  one-fourth  of 

1 A  paper  presented  by  the  author  before  the  International  Electrical  Congress 
at  St.  Louis  in  1904.     Reprinted  from  the  Transactions,  Vol.  I,  pp.  620-634. 
2  See  La  Lumiere  electrique,  1893. 

270 


METHODS  OF  TESTING  ALTERNATORS 


271 


FIG.  18. 


a    pole.      The    system    of    two     machines    is    driven    by  a  meas- 
uring   motor    whose    duty    it    is    to    furnish   the    power    necessary 
for    satisfying    the    losses.       Between    the    two    alternators    AiA2 
(Fig.  1 8)  whose    terminals    are    connected    each    to    each    by    very 
short      couplings     of     negligible      imped- 
ance, a  volt-meter   V  is  connected  across, 
an  ammeter  A  and    a   wattmeter  W  being 
inserted   in    series.     The    figure    is    drawn 
upon  the  supposition    of    two   single -phase 
machines,  but    applies    equally  well  to  the 
case  of    two  similar    three-phase   machines 
coupled  by  their  three  phases,  testing  upon 

a  single  phase,  taking  care  that  the  phases  remain  balanced,  in  spite 
of  the  measuring  instruments. 

Let  U=OB  (Fig.  19)  be  the  difference  of  potential  observed  at 
the  common  terminals,  OA\  and  OA2  the  direction  of  the  generator 
e.m.f.'s  dephased  relatively  to  each  other  by  the  angle  a.  By  sym- 
metry, the  vector  Ob,  which  represents  the  current,  will  also  be  directed 
along  OB,  and  the  line  A±A2  drawn  from  B  perpendicularly  to  U 
and  7,  will  represent  the  double  transverse  reaction  2coLlI.  There 
will  be  a  flow  of  current  between  the  alternators 
without  the  production  of  any  external  power. 
The  current  will  be  in  phase  with  the  E.M.F. 
at  the  terminals  U,  as  if  the  alternators  sup- 
plied a  conducting  system  devoid  of  induct- 
ance ;  the  flux  density  obtained  in  the  armature 
will  be  the  same  in  both  alternators,  since  it 
gives  rise  to  the  same  E.M.F.  at  the  terminals 
U.  The  power  furnished  by  each  will  be 
measured  by  the  wattmeter  W,  and  the  total 

loss  p  will  be  furnished  by  the  method  of  double-weighing,  by  means 
of  the  measuring  motor  which  drives  both  alternators.  The  efficiency 
will  then  be  the  ratio 


To  determine  the  excitations  necessary  for  the  two  alternators 
to  produce  the  condition  above  described,  it  is  sufficient  to  apply  the 
graphic  method  of  two  reactions,  as  follows  (Fig.  19):  from  the  point 


272 


METHODS  OF  CALCULATION 


B  a  perpendicular  BC  is  let  fall  upon  the  straight  line  OAl7  and  the 
condition  is  such  as  if  the  alternator  were  delivering  power  to  an 
inductive  system  according  to  the  total  characteristic  of  excitation 
(Fig.  20).  The  current  /  is  formed  of  two  components:  one  an 
active  component  Oa,  equal  to  the  projection  of  /  upon  OA,  and 

which  gives  the  transverse  reac- 
tion BC  equal  to  a>LIw;  the 
other  component  is  a  reactive 
or  reactive  component  ab,  which 
gives  the  fall  of  potential  to  be 
calculated  on  the  characteristic. 
Let  us  lay  off  on  this  charac- 
teristic (Fig.  20)  an  ordinate 
equal  to  OC,  and  upon  the 
latter  a  segment  CD  equal  to 
the  e.m.f.  lost  by  the  stray 

field.  From  the  point  D  lay  off  horizontally  the  counter  ampere- 
turns  of  the  armature,  and  thus  will  be  obtained  on  the  abscissa 
O'P  the  total  ampere-turns  necessary  for  the  excitation.  The 
ordinate  PPf  corresponding  thereto  will  represent  the  e.m.f. 
E  on  open  circuit  necessary  for  alternator  A\  and  which  will  be, 
in  general,  different  from  the  length  of  OAi  which  was  represented 
in  Fig.  19. 

In  the  same  way  the  e.m.f.  is  determined  which  is  necessary 
for  the  alternator  OA2,  observing  that  for  the  latter  the  sign  of  the 
reactive  armature-reaction  is  changed,  as  well  as  the  sign  of  E=--  A<iQ, 
and  that,  consequently,  the  armature  reaction  remains  demagnetizing, 
so  that  the  geometrical  construction  is  identical.  It  is,  therefore, 
easy  to  recognize  in  advance  the  equal  excitations  to  be  given  to  the 
two  alternators,  in  order  to  satisfy  the  desired  conditions.  It  should 
also  be  determined  at  the  time  of  the  test,  by  means  of  the  wattmeter, 
that  there  is  no  sensible  difference  of  phase  between  the  current  and 
the  e.m.f.  Inversely,  if  this  condition  were  directly  realized  by 
adjusting  the  excitations,  it  would  be  possible  to  deduce  from  an 
examination  of  the  diagram  the  total  armature-reactions  represented 
by  the  abscissa  OP,  indicating  the  total  fall  of  excitation  between 
the  open  circuit  e.m.f.  and  the  e.m.f.  under  load.  In  the  latter 
case,  the  fall  due  to  the  stray  field  is  not  separated  from  that  due  to 
armature  reaction. 

The  same  diagram  gives  immediately  the  value  of  the  transverse 


METHODS  OF  TESTING  ALTERNATORS 


273 


reaction,  since  the  angle  a  is  known  experimentally,  and  the  values 
of  U  and  /  can  be  consequently  measured.     This  gives 


from  which  L'  is  known  as  a  function  of  a,  U  and  7. 

The  same  method  permits  varying  the  angle  a  successively,  and 
repeating  the  operation,  commencing  each  time  with  the  same  voltage 
at  the  terminals  U,  and  thus  tracing  the  entire  characteristic  of  an 
alternator  operating  upon  a  dead  resistance. 

The  above  method  gives  immediately  the  values  of  the  direct 
and  transverse  armature  reactions.  As  to  the  coefficient  of  self-induction 
of  the  stray  field  a>s,  it  may  be  determined  for  any  given  alternator 
by  the  method  indicated  later  on. 

A  test  may  then  be  made  of  the  two  alternators  coupled  together, 
without  any  angular  difference  of  phase  between  them.  The  e.m.f.'s.  E\ 
and  E2  are  then  in  simple  opposition  of  phase,  and  the  difference  EI  —E2 
will  produce  a  resultant  current  which  may  be  regulated  in  strength 
by  regulating  the  difference  of  excitation,  and  which  current  is  dephased 
by  nearly  90°.  The  diagram  is  given  in  Fig.  21,  where  OC\  and  OC2 


o 
FIG.  21. 

are  the  two  internal  e.m.f/s.  The  difference  C\C2  represents  the 
fall  of  potential  due  to  the  impedance  of  the  circuit  of  the  armatures, 
and  which  can  be  decomposed  into  two  rectangular  straight  lines, 
C\F  representing  the  total  ohmic  drop  (Ri+R2}I  due  to  the  current, 
and  C2F  the  total  reactive  drop  in  the  armatures.  Projecting  F  upon 
C\C2j  a  vector  C2F'  is  obtained,  which  differs  but  little  from  C2Ci, 
and  which  represents  the  fall  of  potential  of  the  two  machines  due  to 
direct  reaction.  If  the  characteristic  of  total  excitation  be  then  drawn 
as  in  Fig.  22,  this  drop  will  represent  the  sum  of  the  two  drops  due 
to  the  reactive  current,  of  which  one  CC\  is  positive  and  the  other  CC2 


274 


METHODS  OF  CALCULATION 


is  negative.  It  is  easy  to  mark  these  off  on  the  characteristic.  In- 
versely, the  drops  due  to  the  reactive  current  about  the  point  C  may  be 
deduced,  and  thus  the  coefficient  K  of  direct  reaction.  It  is  sufficient, 
starting  from  the  point  C,  to  trace  two  segments,  CCi,  CC2,  representing 
the  two  drops,  and  to  trace  the  horizontal  C\T\,  C2T2;  thence  the 
abscissas  O/i,  O/2,  which  represent  the  virtually  lost  ampere-turns. 
If  the  segments  CD\,  CD2  are  known,  which  represent  the  e.m  f.'s. 
of  dispersion,  and  if  the  horizontal  straight  lines  are  drawn  through 
DI  and  D2,  the  corresponding  abscissas  t\  and  t2f  permit  of  calcu- 
lating exactly  the  back  ampere-turns,  t\ti  ,  t2t2f,  represented  by  the 
armature,  and  which  should  have  equal  magnitudes. 

Method  No.  2.  Applicable  to  a  Single  Synchronous  Machine 
Operating  upon  an  Actual  Conducting  System.  When  only  one 
alternator  is  available  for  the  test,  it  is  not  possible  to  proceed  so 
conveniently  as  in  the  last  case,  and,  in  particular,  the  plan  of  testing 
with  variable  angles  of  coupling  must  be  given  up. 

A  similar  test  to  that  which  we  have  indicated  above  can,  how- 
ever, be  made  by  driving  the  alternator  on  open  circuit  as  a  syn- 
chronous motor  supplied  from  the  conducting  system  on  which  it 
is  to  be  employed  (supposing  the  factory  to  have  other  alternators 
already  installed)  or  by  a  current  furnished  from  some  other  alter- 
nator of  equivalent  power.  The  alternator,  or  alternators,  serving 
as  the  source,  will  then  be  excited  in  such  a  manner  as  always  to 
maintain  the  voltage  constant  at  the  terminals  of  the  alternator  under 
test  which  operates  as  a  synchronous  motor;  this  voltage  being  the 

normal  voltage  of  operation,  the  excita- 
tion of  the  motor  is  to  be  varied,  as  if  it 
were  desired  to  obtain  the  "  V-curve  "  of 
constant  voltage.  The  latter  gives  by 
its  minimum  ordinate  AB  (Fig.  23)  the 
value  of  the  ohmic  losses  (at  current 
/o),  and  the  indication  of  the  condi- 
tion of  excitation  OA  corresponding  to 
a  power-factor  equal  to  unity  (cos  <f>=  i) 
— at  least  on  the  hypothesis  that  the 
effects  of  harmonics  in  the  e.m.f. 

are  inconsiderable.  For  any  other  excitation  Oa,  the  strength  of  the 
reactive  current  may  be  obtained  by  constructing  upon  ab  a  triangle 
of  which  the  angle  at  b  is  given  by  the  wattmeter.  The  side  bd=Iw, 
differs  little  from  BA,  that  is  to  say,  from  the  active  current  on 


FIG.  23. 


METHODS  OF  TESTING  ALTERNATORS 


275 


open  circuit;  ad  then  represents  the  reactive  current  Id-  The  value 
of  the  reactive  current  delivered  or  received  by  the  motor  may  then 
be  deduced  from  the  V-curve  for  all  values  of  excitation.  If  reference 
is  made  to  the  characteristic  in  Fig.  24,  on  which  OC  represents  the 
normal  e.m.f.  at  the  terminals,  the  knowledge  of  Aa  gives  for  each 
value  of  the  current  7^  the  corresponding  value  of  the  total  lost  ampere- 
turns  OF=Aa.  A  curve  of  these  ampere -turns  may  then  be  drawn 
as  a  function  of  7rf,  and  the  total  fall  of  potential  CD  thus  deduced 
from  the  chart  for  every  value  of  the  armature  current.  This  method 
does  not  separate  the  reaction  into  two  parts  according  to  theory, 
but  it  gives  exactly  the  required  result  which  makes  it  possible  to 
calculate  the  fall  of  potential  for  each  value  of  the  current  and  of 
the  phase  with  respect  to  the  mean  e.m.f.  CD  which  the  armature 
designed  to  supply. 


'  To  complete  this  indication  it  is  sufficient  to  know  the  transverse 
reaction.  This  may  be  determined  readly  enough  by  the  same 
experiment  if  care  be  taken  to  measure  the  phase  angle,  <£>,  of  the  cur- 
rent with  respect  to  U,  the  mean  e.m.f.  at  the  terminals.  Let  us,  then, 
draw  (Fig.  25)  the  synchronous  motor  diagram  with  OA  to  represent 
the  internal  induced  e.m.f.,  E,  and  OB  the  mean  voltage,  U,  at 
the  terminals;  then  the  geometrical  difference,  AB,  will  represent 
the  fall  of  potential  due  to  impedance,  which  may  be  resolved  into 
two  vectors,  one  of  which  (BH)  represents  the  ohmic  drop,  and  the 
other  (HA)  the  fall  of  potential  wL'I  calculated  as  a  function  of 
the  transverse  reaction.  Knowledge  of  the  angle  <j>  and  the  current 
strength  7  enables  the  point  77,  and  also  the  direction-line  HA,  to 
be  determined.  Knowing,  from  the  experiments  set  forth  below, 
the  angle  of  lag,  </>,  of  the  internal  e.m.f.,  E,  (whose  direction,  OA, 
can  thus  be  drawn)  it  is  only  necessary  to  take  the  point  of  intersection 
of  OA  with  the  direction-line  HA  to  locate  the  point  A,  and,  conse- 
quently, to  be  able  to  complete  the  triangle  OH  A.  The  length  of 


276  METHODS  OF  CALCULATION 

HA  gives,  at  once,  the  value  of  the  transverse  inductance  L '.  The 
same  determination  can  therefore  be  made  over  again  with  increasing 
values  of  </>,  by  simply  increasing  the  excitation  each  time.  The 
greater  the  lag-angle,  </>,  the  more  accurately  L'  will  be  determined. 

It  may  be  noted,  moreover,  (Fig.  250),  that  if  a  line  (BD)  is  drawn 
from  B  perpendicular  to  OA,  the  portion  intercepted  (OD)  corre- 
sponds to  the  e.m.f.  e  obtained  with  no-load.  The  difference 
DA'=E—e  therefore  gives,  in  volts,  the  value  of  the  armature- 
reaction  produced  by  the  reactive  component  of  the  output  current. 

When  the  angle  at  A  (Fig.  250)  is  very  small,  we  have  substantially, 


Let  Zi 

If  we  let  Ld  =  the  direct  reaction,  then,  for  a  very  small  variation 
of  the  excitation  we  can  write 


whence  AAf  =  u(Lt  —  Ld)Id. 

Instead  of  evaluating  in  the  preceding  test  the  active  current  Iw 
in  order  to  deduce  the  reactive  current  Id,  it  would  be  easy,  if  a 
measuring  motor  were  at  hand,  to  make  this  deliver  directly  to  the 
shaft  of  the  alternator  operating  as  a  synchronous  motor  the  necessary 
power  for  driving  the  motor,  in  such  a  manner  that  AB  on  the  V-curve 
(Fig.  23)  becomes  nil.  But  the  same  result  may  be  obtained  yet  more 
easily  when  a  steam  alternator  unit  has  to  be  tested,  by  admitting 
to  the  engine  just  enough  steam  to  satisfy  the  losses  both  of  the  engine 
and  alternator,  so  that  the  alternator  only  receives  a  reactive  current. 
With  this  object,  the  steam  admission  may  be  regulated  in  such  a 
manner  that  the  alternator,  excited  so  as  to  give  on  open  circuit  the 
normal  e.m.f.  U  of  the  system,  runs  idly  in  synchronism;  then  no 
change  is  made  either  in  steam  admission  or  in  the  pressure,  and  the 
operations  are  conducted  entirely  on  the  electric  side  of  the  alternator, 
connecting  this  with  the  system  and  varying  its  excitation  so  as  to 
develop  the  V-curve.  It  is  possible  to  measure  in  advance  the  elec- 
tric power  necessary  to  drive  the  alternator  and  its  steam  engine  on 
open  circuit,  and  thus  to  deduce  the  total  losses  on  open  circuit. 
The  power  wasted  may  thus  be  measured  by  the  steam-engine 
indicator-diagram,  which  permits  of  determining  the  constant  of  the 
curve  of  steam  consumption  as  a  function  of  the  power  produced 


METHODS  OF  TESTING  ALTERNATORS 


277 


(a  curve  which  generally  is  nearly  a  simple  straight  line);  it  is  also 
possible  to  separate  the  mechanical  losses,  first  making  this  test  without 
field-excitation,  and  then  exciting  the  field-magnets  so  as  to  obtain 
the  normal  e.m.f.1 

Summing  up,  this  test  permits  of  determining,  with  a  fair  approx- 
imation, the  losses  on  open  circuit  and  then  on  load,  without  being 
obliged  to  actually  develop  these  losses  by  full -load  in  the  alternator 
as  well  as  the  corresponding  heatings. 

From  an  electrical  point  of  view  the  same  test  permits  of  deter- 
mining the  total  values  of  the  direct  reaction  as  a  function  of  the 
reactive  current,  and  the  constant  Z/  of  the  transverse  reaction.  If 
it  is  desired  to  analyze  these  phenomena  more  completely,  the  value 
of  cos  may  be  determined  as  follows: 

Determination  of  ojs.  For  this  purpose  the  following  consider- 
ations are  made  use  of  (the  basis  of  which  is  the  method  of  calculation 
of  the  short-circuit  current  given  by  Kapp,  which  does  not  lend  itself 
to  experimental  verification): 

Let  ONM  be  the  characteristic  of  excitation;  that  is  to  say,  the 
curve  of  the  armature  e.m.f.  as  a  function  of  the  exciting  ampere- 
turns.  Suppose  the  excitation  to  be  constant  and  equal  to  OA 
ampere-turns,  giving  an  e.m.f.  represented  by  AM.  Suppose  the 
reactance  cos  sought  for  to  be  known 
and  also  the  short-circuit  current  7^. 
If  a  point  P  is  taken  on  the  curve 
whose  ordinate  is  equal  to  coslcc,  the 
corresponding  segment  of  the  abscissa 
AH  will  measure  the  back  ampere- 
turns  of  the  armature  KNI^.  If  the 
calculated  values  of  cos  and  of  K  are 
taken,  the  direction  of  the  straight  line 
AP  is  known,  and  the  line  may  be 
traced  from  which  the  value  of  7^  may 

be  deduced.  This  is  the  construction  of  Kapp.  We  shall  take  up,  on 
the  contrary,  the  inverse  problem,  supposing  Icc  determined  by  experi- 
ment and  seeking  to  deduce  from  it  the  two  constants  K  and  cos. 

The  following  observation  may  then  be  made.  If  the  segment 
AH  be  taken  as  the  measure  of  the  short-circuit  current  7^  and  a 
segment  AB  as  the  measure  of  any  other  reactive  current  Id  to  the 

The  total  efficiency  of  the  engine  and  alternator  unit  may  also  be  obtained 
later  at  load  by  means  of  the  engine  diagram. 


Ampere-turns 


FIG.  26. 


278  METHODS  OF  CALCULATION 

same  scale  of  volts  at  the  terminals  corresponding  to  this  reactive 
delivery,  this  segment  will  be  equal  to  the  ordinate  QN=  U,  taken 
between  the  curve  and  the  straight  lin,e  AP.  In  fact,  the  back  ampere- 
turns  will  then  be  equal  to  AB,  and  the  loss  of  voltage  by  dispersion 
equal  to  BQ,  by  reason  of  similar  triangles.  Besides,  the  point  Q 
divides  the  straight  line  AP  in  the  ratio  of  the  current  Id  to  the  short- 
circuit  current  Icc. 

Thus  arranged,  suppose  that  any  reactive  current  Id  be  taken 
experimentally  at  the  corresponding .  e,m.f.  £/;  the  point  Q  will  be 
determined  by  these  two  conditions:  its  vertical  distance  from  the 
curve  of  excitation  is  equal  to  U  and  its  radial  distance  from  the 

point  A  is  equal  to  —  Y.AP. 

Ice  .,  ,: 

The  point  is  found,  therefore,  at  the  intersection  of  two  new 
curves  that  are  easily  drawn:  a  curve  parallel  to  ONM  traced  at 
a  vertical  distance  U  below  the  former,  and  a  curve  homothetic 
to  the  curve  of  excitation  with  respect  to  the  point  A,  with  homo- 
thetic ratio  — .  These  two  curves  are  parallel  in  their  rectangular 

ec  ;      .  , 

parts,  and  separate  as  much  one  from  the  other  as  the  point  M  is 
selected  further  beyond  the  bend  in  the  characteristic.  They  would 
coincide  if  the  point  M  were  below  the  bend.  The  experiment  should 
therefore  Lbe  made  with  an  excitation  OA  sufficient  clearly  to  pass 
the-bendj. 

The  test  js  made  by  causing  the  alternator  to  operate  first  on 
short-circuit,  and  then  upon  a  reactance-coil  having  an  open  mag- 
netic circuit,  of  uppn  an  under-excited  synchronous  motor,  giving 
cos0  less  Q/2QJ.  that  is  to  say,  a  current  almost  entirely  reactive. 

It  is  '  understood  that  if  not  only  one,  but  also  several,  reactive 
circuits  are  tried,  the  straight  line  PQ  will  be  still  better  determined 
thereby,  and  consequently  cos  will  be  known  with  correspondingly 
greater  precision. 

Analogies  between  this  Method  and  that  of  Potier-Behrend. 
,It  is  possible  also  to  follow  a  somewhat  different  course,  by  causing 
the  alternator  under  test  to  operate  upon  an  inductive  circuit  with 
;cqs<£  nearly  o  (for  example,  upon  a  synchronous  motor  driven  by 
-a  ; motor  adjusted  in,  a  manner  to  produce  just  the  power  consumed 
rat;  light  load),  and  by  varying  the  excitation  of  the  alternator;  in 
T this  manner  a  constant  reactive-current  curve  is  obtained,  from  which 
'may  be  deduced,  by  the  method  of  M.  Potier^the  coefficient,  of  the 


METHODS  OF  TESTING  ALTERNATORS  279 

back  ampere-turns,  and  the  coefficient  of  the  stray  field,  but  which 
is  not  directly  applicable  to  the  ordinary  problem  of  the  calculation 
of  the  ampere-turns  necessary  for  constant  voltage. 

This  method  is  one  of  those  which  was  devised  by  Mr.  B.  A. 
Behrend,  who  recently  published  a  number  of  applications  of  it.  It 
is  wholly  different  from  the  preceding,  in  which  the  voltage  is  main- 
tained constant  at  the  terminals  instead  of  the  current.  The  same 
author  employs  also  another  method,  which  consists  in  dividing  the 
field  magnets  into  two  equal  parts  to  which  are  given  exciting  currents 
of  opposite  sign  and  slightly  different  strength,  so  as  to  develop  in 
the  armature  a  certain  current,  which  is  necessarily  reactive  while 
developing  a  mean  flux-density  sensibly  equal  to  that  in  normal 
operation.  This  test  may  appear  equivalent  to  that  which  is  obtained 
by  means  of  two  alternators  mechanically  coupled  and  set  at  opposite 
phase  (see  preceding  Method  No.  i),  with  the  simplification,  however, 
that  only  one  alternator  is  employed;  but  as  the  current  thus  obtained 
is  only  a  reactive  current,  the  conditions  are  not  identical  with  those 
of  the  preceding  method,  but  only  equivalent  to  method  No.  2  above 
described  (see  the  case  of  the  alternator  operating  as  a  synchronous 
motor  on  no-load).  It  is  not,  besides,  sufficiently  rigorous  except 
for  alternators  having  a  large  number  of  poles,  because  when  there 
are  but  few  poles  the  presence  of  two  poles  of  the  same  sign  side  by 
side  upon  the  field  magnet  at  two  points  on  the  latter  vvould  seem 
to  modify  notably  the  conditions  of  the  magnetic  circuit.  Conse- 
quently, Method  No.  i  is  preferable  when  it  can  be 
employed. 

Method  No.  3.     For  the  Determination  of  Trans- 
verse Reaction  (coefficient  L).     Besides  the  preced- 
ing methods,  several  others  may  also  be  pointed  out 
which  are  very  simple  for  the  determination  of  the 
transverse  reaction,  and  which   avoid   the  objection 
so  often  made  to  the  diagram  in  which  this  reaction 
appears.       Moreover,    the    oscillograph    (or    ondo- 
graph)  permits  the    real  angle    of   dephasing    ^  to 
be  measured  between  the  internal  e.m.f.  E  (Fig.  27)  and  the  current,  and 
gives  immediately,  in  consequence,  the  value  of  L,  when  the  alternator 
operates  upon  a  non-inductive  resistance  </>=o,  by  the  equation 
.     coLI  +  U  sin  (h      a)L 


280  METHODS  OF  CALCULATION 

from  which 

/     m 

wL=  I  'f~^~~Y )  tan  0. 

With  the  oscillograph,  for  example,  it  suffices  to  place  upon  the 
shaft,  or  upon  one  of  the  pole  pieces  of  the  rotating  field-magnet 
(Fig.  60),  a  contact  segment  S  upon  which  rub  two  brushes  BB', 
connected  in  series  with  a  battery,  one  of  the  oscillographs  O  of  a 
double-oscillograph  apparatus,  and  the  electromagnet  e  which  controls 
the  release  of  the  shutter.  The  contact  segment  5,  insulated  from  the 
base,  closes  the  circuit  upon  its  passage  below  the  brushes,  and  opens 
it  again  at  the  precise  moment  when  the  brush  B  leaves  it.  A  segment 
is  selected  large  enough  in  order  that  the  shutter  shall  have  time  to 
open  before  the  rupture  of  the  circuit,  in  order  that  the  latter  may  then 
be  photographed  upon  the  plate  in  the  form  of  a  vertical  line  inter- 
secting the  straight-line  zero  furnished  by  one  of  the  oscillographs. 


s 

FIG.  28. 

The  second  oscillograph  of  the  system,  O',  is  connected  to  the  terminals 
of  the  alternator  and  serves  to  register  the  e.m.f.  U  at  the  terminals. 
Two  experiments  are  made  on  two  different  plates,  or  upon  the  same 
plate  after  having  slightly  displaced  the  zero-line  produced  by  the 
oscillograph  O,  so  as  to  distinguish  the  two  different  records.  The 
e.m.f.  at  open  circuit  is  then  marked  upon  it,  and  next  the  e.m.f. 
when  the  circuit  is  closed  upon  a  dead  resistance,  at  the  same  time 
that  the  current  I  delivered  thereto  is  measured.  In  these  two  tests 
the  distance  is  measured  from  the  zero  of  the  curve  of  e.m.f.  to  the 
point  of  intersection  by  the  vertical  with  the  zero-line.  The  difference 
between  these  two  lengths  thus  measured  determines  the  displacement  of 
phase  of  the  e.m.f.  at  the  terminals,  that  is  to  say,  the  angle  of  dephasing 
sought  (taking  for  the  value  of  271  the  length  of  a  period  measured  upon 
the  plate,  and  taking  the  ratio  of  the  measured  retardation  to  this 
length).  It  is  sufficient  to  take  the  values  of  </>,  of  /,  and  of  U  in  the 
preceding  formula  in  order  to  determine  the  transverse  reaction  L. 
Figs.  130,  140,  150  and  16  (Chapter  I)  represent  an  example  of  the 


METHODS  OF  TESTING  ALTERNATORS  281 

determination  of  L  made  by  this  method  in  the  laboratory  of  the 
Societe  Sautter-Harle  &  Company  of  Paris,  which  employs  with  success 
the  methods  described  in  this  note.  The  curves  of  the  first  three 
figures  have  been  obtained  by  means  of  the  author's  oscillograph  at 
the  terminals  of  an  alternator  of  350  K.W.,  and  represent  the  periodic 
curves  of  the  difference  of  potential  between  these  terminals  when 
the  alternator  works  on  open  circuit,  then  when  delivering  62 
amperes,  and  finally  102  amperes,  its  normal  load.  The  effective 
value  of  the  e.m.f.  was  1155  volts  in  star.  The  vertical  lines 
represent  the  tracings  produced  by  the  contact  of  the  contact-maker 
S.  Figs,  i $a  and  150  are  printed  upon  one  and  the  same  sheet  of 
paper,  by  causing  the  vertical  lines  to  coincide,  from  which  Fig.  16 
is  produced,  which  shows  clearly  in  evidence  the  dephasing  between 
the  internal  e.m.f.  E,  and  the  pressure  at  terminals  £7.  The 
dephasing  reaches  24°  30'  in  the  test  with  102  amperes,  and  19°  in 
the  test  with  62  amperes.  In  applying  the  preceding  formula  (with 
r=2.5  ohms)  the  values  are  obtained: 

^£=(2.5X62+1155)   Xo.344-y.26  ohms, 
<wL=  (2. 5X102 +1155)  Xo.566=  7.81  ohms, 

which  differ  little  (and  would  perhaps  be  equal  if  greater  precision 
had  been  taken  in  the  measurement  of  the  phase-difference  between  the 
two  curves).  In  practice  one  would  take  the  mean  value  7.5  ohms. 

It  is  evident  that  it  is  easy  thus  to  determine  the  constant  of 
distortion,  and  this  justifies  the  employment  of  the  theory  of  the  two 
reactions  rather  than  the  rough  method  of  the  "  curve  of  short-circuit," 
which  combines  together  the  two  often  very  different  reactions. 

The  ondograph  gives  the  same  result,  if  one  marks  off  successively 
upon  the  same  sheet  of  paper  the  e.m.f.  upon  open  circuit  and  then 
that  on  closed  circuit,  and  each  time  the  mark  obtained  when  the 
apparatus  is  traversed  by  the  current  coming  from  the  brushes  BB'. 
It  is  possible  to  dispense  with  making  this  mark,  if  the  ondograph 
is  driven,  not  by  a  synchronous  motor,  but  by  a  flexible  coupling 
connected  mechanically  to  the  shaft  of  the  alternator  (with  the  inter- 
position of  gear  wheels,  as  in  the  recording  mechanism  of  Francke). 

Moreover,  even  in  the  absence  of  the  preceding  analyzing  appa- 
ratus, one  can  approximately  obtain  the  transverse  reaction,  or  rather 
its  ratio  to  the  direct  reaction,  with  unsaturated  field-magnets,  by 
sending,  as  M.  Herdt  has  already  suggested,  into  the  alternator  arma- 


282  METHODS  OF  CALCULATION 

ture  at  rest,  alternating  currents  derived  from  any  available  external 
source,  and  measuring  the  apparent  self-induction  by  the  method 
of  Joubert  for  the  two  characteristic  positions  of  the  field-magnet 
poles.  The  simplest  method  applicable  to  a  star-wound  alternator 
consists  of  sending  an  alternating  current  7  through  one  of  the  phases 
which  divides  between  the  two  other  phases,  starting  at  the  neutral 
point,  and  which  is  received  at  its  emergence  by  a  connection  applied 
to  the  two  terminals  A2A^.  Fig.  29  represents  the  connections:  a 
is  an  auxiliary  alternator  of  which  the  excitation  is  adjusted  at  will 
in  order  to  vary  the  current  7.  It  is  easy  to  see  that  an  alternator 
thus  traversed  by  a  current  is  placed  in  the  same  conditions  as  if  it 
were5  fed  by  three-phase  currents  at  the  moment  when  one  of  these 
currents  passed  through  its  maximum.  This  connection  being  made, 
the  field-magnet  is  arranged  so  that  the 
armature  poles  have  their  axes  directed  in 
a  line  with  those  of  the  field-magnet  poles, 
and  then  in  such  a  manner  that  these  axes 
are  directed  midway  between  the  field- 
magnet  poles^  In  each  case  the  self-induc- 
tion developed  by  the  current  7  is  measured 
according  to  the  method  of  Joubert;  the 
FlG  ratio  of  the  two  measures  is  that  of  the 

direct   self-induction  to   the   transverse   self- 

induction,  or  of  the  coefficients  K  and  Kt  applicable  to  these  two 
reactions.  If  the  difference  of  potential  u  is  measured  between  A 

and  the  center  of  the  star  O,  the  ratio  —  represents  the  impedance 

of  one  phase.  From  this  the  inductance  may  be  determined,  since 
the  resistance  of  one  phase  is  known.  This  measure  made  in  the 
second  position  indicated  above,  gives,  therefore,  the  precise  and 
approximate  value  of  the  transverse  inductance  sought.  It  must  be 
remembered  that  this  inductance  also  comprises  the  inductance  of 
dispersion;  therefore  has  the  value 


calling  a)X  the  transverse   reaction  properly  so  called,   not  including 
the  dispersion  cus. 

Repeating  the  experiment  for  various  values  of  the  current,  the 
constancy  of  the  coefficient  L  may  be  determined. 


APPENDIX  A 


IF  Eq.  (A)  on  page  16  be  solved  completely  for  i,  the  solution 
will  include  a  transient  exponential  term  depending  upon  the  point 
of  the  E.M.F.  cycle  at  which  the  circuit  is  closed.  It  is  a  well-known 
fact  that  this  transient  term  becomes  negligible  a  short  time  after 
the  circuit  is  closed,  and  that  when  the  impressed  E.M.F.  is  sinu- 
soidal the  current  i  will  settle  down  to  a  sinusoidal  form. 

In  the  present  case  both  of  the  active  E.M.F. 's  (e\  and  62)  are 
assumed  sinusoidal  and  of  the  same  frequency;  their  resultant  is  there- 
fore sinusoidal  as  will  be  the  current  produced  thereby.  Then  all  the 
variables  involved  are  sinusoidal  and  may  be  represented  by  vectors 
in  such  a  manner  as  to  indicate  clearly  their  phase-relations. 


x' 


o 

FIG.  A 


Referring  to  Fig.  A,  draw  OC,  to  designate  by  its  length  and 
direction  the  phase  and  magnitude  of  the  internal  or  induced  E.M.F. 
EI  of  the  generator,  and  OA  to  designate  in  a  similar  manner  the 
internal  or  induced  E.M.F.  £2  of  the  motor.  Then  the  resultant 
or  vector-sum  of  E\  and  £2  will  be  E,  designated  by  the  line  OB. 

The  horizontal  line  OX  is  taken  arbitrarily  as  the  zero  of  phase,1 

1  In  Fig.  A  the  instantaneous  values  of  the  various  variables  (e.g.,  e\  and  e2) 
are  given  by  the  vertical  projections  of  the  corresponding  vectors  (multiplied 
by  \/~2),  as  the  latter  revolve  counter-clockwise  at  constant  angular  velocity, 
o>  =  2TCW,  where  n  =  cycles  per  second.  The  diagram  is  shown  at  the  instant  t  =o. 

283 


284  APPENDIX  A 


and  the    Z.COX=  £AOX=-.    This  corresponds  to  the  equations 
for  e\  and  62  on  page  16,  which  show  that  when  t=o,  e\  has  a  phase 

A 

H —  and  e2  falls  short  of  exact  opposition  (180°)  by  the  same  angle 

A 

— ,  where  6  is  the  angle  between  E\  and  E2  projected  backwards. 

E2  may  be  said  to  lag  behind  E\  by  i8o°+0,  or  to  lead  E\  by  180°— 6. 

The  current  produced  in  the  circuit  of  the  two  armatures  will 
be,  I=E/Z  and  will  lag  behind  E  by  the  angle  y  whose  tangent 
is  X+R-,  where  R,  X  and  Z,  (  =  VR2+X2),  are  respectively,  the 
resistance,  the  reactance,  and  the  impedance  of  the  whole  circuit, 
including  the  impedance  of  the  connecting  line  and  the  synchronous 
impedance  of  the  two  armatures;  and  X=2i<:nL  =  tt>L}  where  L 
is  the  inductance  of  the  whole  circuit. 

By  trigonometry 


-2EiE2  cos  6, 
and  since 


we  have 


E=/Z,    and    /=f , 


E  _VEi2+E22-2EiE2  cos  6 
Z~  Z 


and  the  current-phase  (referred  to  OX)  is  T—  p—  y. 

These  two  relations  stated  algebraically,  give  as  the  instantaneous 
value  of  i, 

V^VEi2+E22-  2EiE2  cos  0         ,   .  v 


*= 


which  is  the  same  as  the  value  of  i  given  at  the  top  of  page  17. 

The  other  forms  of  equations  for  i  and  /  given  on  page  17  are 
deducible  from  the  diagram  of  Fig.  A. 


APPENDIX  B 


DATA  CONCERNING  SYNCHRONOUS  MOTORS  OF  AMERICAN 
MANUFACTURE 

THE  Tables  A  and  B,  given  herein  below,  contain  data  charac- 
teristic of  good  American  practice  in  the  manufacture  of  synchro- 
nous motors.  In  comparing  these  data  with  those  characteristic 
of  good  European  practice,  as  given  in  Part  I,  on  page  5,  due  allow- 
ance should  be  made  for  the  rating  conditions  in  the  two  cases. 
Thus,  the  normal  frequency  is  60  for  the  American  motors,  and 
40  for  the  European  motors;  and  the  kilo  volt-amperes  per  horse- 
power are  746  in  America  and  736  in  Europe.  The  power-factor 
assumed  as  a  basis  for  rating  is  equal  to  unity  for  American  motors. 
It  is  presumably  the  same,  though  not  stated,  for  European  motors. 

TABLE     A.— SINGLE-PHASE     AND  '  THREE-PHASE     SYNCHRONOUS 
MOTORS  OF  AMERICAN  MANUFACTURE 


i>> 

E- 

No.  of  Poles. 

1 

Pulley. 

43 

0> 

Single-phase. 

Three-phase. 

-S| 

QM 

Face  in 
Inches. 

a 

"o 

l> 

M 

H 

P 

«'H 

AT* 

4 

1800 

8 

3-5 

715 

6.5 

79 

56 

6-5 

10 

45-6 

10 

AT* 

4 

1800 

10 

4-5 

1070 

13- 

82.5 

1  08 

12 

20 

89-2 

17-5 

AT* 

4 

1800 

12 

5-5 

1535 

22.5 

83 

185 

2O 

35 

155 

30 

ATBf 

6 

1  200 

IS 

8 

2250 

32.5 

85 

265 

30 

50 

220 

45 

ATB| 

6 

1  200 

16 

n 

2580 

43-5 

87-5 

335 

37-5 

67 

287 

55 

ATBJ 

6 

1  200 

16 

14 

3120 

65 

89 

495 

55 

TOO 

422 

80 

ATBt 

8 

900 

21 

17 

4380 

87 

89-5 

660 

75 

134 

565 

no 

ATBJ 

10 

720 

26.5 

23 

6630 

130 

90 

980 

no 

200 

840 

160 

*  Self-excited. 

t  Separately  excited. 

t  Has  direct-connected  exciter. 


285 


2X<» 


AIM'MMMX     It 


TABU-;     H.     SINCil.KI'HASK     AND     TIIUKK  IMIASK     SYNCHRONOUS 
MOTORS   OK   AMKRICAN    MANUIA(   II  I  I 


0-  z 

<:    .1 
o    <; 

<;  •/ 

c:    o 

(i     M 

G-I3 


i  100 

I20O 

Ooo 

poo 

000 

600 
600 


P 


1  1 
ifl 

90 

90 
90 
30 
30 


i 


1.10" 
I0.|o 


4150 


6500  1 20 

8OOO  131 


170 


41'' 

58-5 

89.5 


85 
87.9 

87.3 
88.0 

r,o   o 
90-5 


*>' 


189 
318 

445 

ny, 

910 

QtO 
1979 


KLW.of 
Tnns. 


Thr«t-phM«, 


3 
36 

(n 
91.  S 

Z93 

i  ;-w, 
186 
250 


158 

363 

',01 

596 

787 

787 

1050 


89-3 

o« 

91 

9Z.4 
93-5 
93-o 

o',    V 


All  these  maehines  have  rotating  poles  and  open  slot  stationary 
a  in  i.i  hues.  'The  held  pole:,  air  equipped,  in  add  it  inn  In  the  standard 
held  <oil:.,  uiih  a  :,hoii  (limited  winding  similar  to  an  induction 
inolot,  vvliK  h  is  tllili/ed  to  make  I  he  machine  sell  starling  as  .in 
iiidmlioii  inoloi ,  .ind  also  as  a  "damper"  to  prevent  "hunting" 
when  running  in  synchronism.  When  starling,  polyphase  machines 
will  develop  Irom  .'o  to  '/'  |)rl  <  ent  ol  their  normal  torque  with 
Iron  100  to  250  per  cent  of  the  normal  input  in  kilovolt-ampcres. 
The  single-phase  machines  arc  not  sell-starting. 


APPENDIX  C 


i;si<;  OK  SYNCHRONOUS  MOTORS  KOK  IMPROVING 
POWER-FACTOR  IN  AMERICA 

IT  IB  well  known  that  the  power-factor  of  a  system  supplying 
current  to  rotary  converter!  at  various  substation**  can  be  mate- 
rially improved  by  over-exciting  the  field  of  the  rotary  converters. 
In  Home  cases  this  benefit  was  considered  of  enough  importance  to 
warrant  the  use  of  rotary  converters  which  are  shunt-wound  only  and 
consequently  can  produce  no  compounding  effect. 

In  a  motor-generator  set  composed  of  a  synchronous  motor 
driving  a  continuous  current  generator  the  compensating  effect  due 
the  over-excitation  of  the  synchronous  motor-element  is  retained 
without  sacrificing  the  compounding  effect  of  the  combination. 
This  is  one  reason  why  motor-generator  sets  have  been  preferred 
in  many  cases  to  rotary  converters. 

Owing  to  the  fact  that  the  compensating  effect  is  the  same  as 
that  produced  by  a  condenser  the  synchronous  motor  producing  the 
"  condenser  effect "  is  often  called  a  "  synchronous  condenser  "  or 
a  "  rotary  condenser,"  and  also  a  "  synchronous  compensator/' 
The  Standards  Committee  of  the  American  Institute  of  Electrical 
Engineers  gives  the  preference  to  another  term,  "  phase-modifier/' 
as  appears  from  the  following  definition  given  in  the  "  Standardiza- 
tion Rules  "  : 

"  A  Synchronous  Phase-modifier,  sometimes  called  a  Synchronous 

Conrlrir.i-r,    i;   ;i   :.ynr}ironous   motor,    running   i-ilhrr    i<ll<-   or    wnlrr 

load,  whose  field-excitation  may  be  varied  so  as  to  modify  the  power* 
factor  of  the  circuit,  or  through  such  modification  to  influence  the 
voltage  of  the  circuit." 

287 


288  APPENDIX  C 

The  practice  of  designing  the  synchronous  motor  portion  of 
motor-generator  sets  with  special  reference  to  their  use  as  "  synchro- 
nous condensers  "  has  become  quite  general  in  America.  Many 
articles  and  papers  have  appeared  on  the  subject.  A  partial  list 
of  the  authors  is  given  at  the  end  of  the  book. 

The  following  item,  taken  from  The  Electrical  World,  refers  to 
an  interesting  case  where  synchronous  condensers  have  been  used 
to  great  advantage: 

(From  the  Electrical  World,  March  g,  1912,  p.  548) 

Overrating  of  Motor-generator  Sets  to  Improve  Power-factor. 
All  motor-generator  sets  installed  by  the  Detroit  Edison  Company 
are  now  specified  with  their  motor-elements  sufficiently  larger  than 
the  generators  to  insure  that,  fully  loaded,  the  motors  can  be  operated 
at  a  leading  power-factor  of  80  per  cent  to  improve  the  power-factor 
of  the  system.  Operators  are  instructed  to  increase  the  field-excita- 
tion of  their  machines  until  the  motors  take  full  load  current  at  the 
leading  power-factor.  As  the  result  of  this  practice,  power-factors 
approaching  within  97  per  cent  of  unity  have  been  observed  at  the 
bus  of  the  generating  plant. 

The  Detroit  Edison  Company,  through  its  Engineering  Depart- 
ment, has  kindly  placed  at  the  disposal  of  the  Translator  the  follow- 
ing detailed  information  regarding  this  case: 

(From  the  Engineering  Department  of  the  Detroit  Edison  Co.) 

The  motor-generator  sets  referred  to  in  the  above  item  are 
synchronous  machines  installed  in  the  substations.  They  are  sup- 
plied with  4600  volt  A.  C.,  and  deliver  600  volt  D.  C.  for  railway, 
and  250  volt  D.  C.  for  the  lighting,  service.  All  are  over-excited 
more  or  less;  and  the  station  operators  are  instructed  to  adjust  the 
field  so  that  the  motors  always  take  the  full  load  current  at  a  leading 
power-factor.  This  power-factor  varies  from  50  to  95  per  cent. 

The  total  induction-motor  load  on  the  systems  is  about  19,000  KW. 
with  a  power-factor  varying  from  60  to  85  per  cent,  lagging.  The 
lagging  current  of  such  a  load  would  lower  the  power-factor  at  the 
power-house  bus  considerably,  but  by  means  of  the  over-excited 
machines  in  the  substations  a  leading  current  is  provided  to  counter- 
act the  lagging  one,  and  a  very  good  power-factor  is  thus  main- 
tained. 


APPENDIX  C 


289 


The  extent  to  which  the  power-factor  at  the  power-house  bus 
is  affected  by  the  over-excitation  of  the  synchronous  motors  is  best 
shown  by  a  vector  diagram. 

This  was  figured  from  simultaneous  readings  on  the  turbo- 
generators at  the  power-house,  and  the  synchronous  motors  at  the 
substations.  The  average  power-factor  of  the  turbo-generators  with 
a  load  of  31,100  KW.,  under  normal  running  conditions,  is  about 
95  per  cent,  lagging.  The  reactive  component  of  this  is  9750  K.V.A., 
lagging.  The  total  load  taken  by  the  synchronous  motors  is  10,735 


12100  K.W.-Bailway  and  Lighting  Load 


18474-Probable  Total  of 
Reactive  K.V.A.  Without 
Synchronous  Motora 


KW.  with  a  power-factor  varying  from  50  to  95  per  cent  leading, 
and  the  summation  of  all  its  reactive  components  at  the  various 
power-factors  is  8724  K.W.A.  This  leading  component  counteracts 
an  equal  lagging  one,  leaving  a  resultant  reactive  lagging  component  of 
9750  K.V.A.  on  the  whole  system,  corresponding  to  a  power-factor  of 
95  per  cent.  Without  this  leading  component  the  lagging  component 
would  be  18,474  K.V.A.,  which  would  mean  a  power-factor  of  85 
per  cent.  Hence  it  is  seen  that  a  10  per  cent  rise  in  power-factor 
is  obtained  by  over-excitation  of  the  synchronous  motors  in  our 
substations. 

Regarding  the  overrating  of  the  motor-ends  of  the  motor- 
generator  sets,  the  following  two  examples  of  rating  might  be 
mentioned: 


290  APPENDIX  C 

MACHINE  No.  i.     (Installed  under  the  old  system.) 
Motor:         Class,  10-530-720. 

Volts,  4400. 

Amperes,  73  per  phase. 
Exciter:       Class,  6-7-720. 

Volts,  125. 

Amperes,  56. 

Generator:  500  KW.  direct-connected. 
MACHINE  No.  2.     (Installed  under  the  new  system.) 
Motor:         Class,  10-700-720. 

Volts,  4400. 

Amperes,  92. 

Power-factor,  0.80. 
Exciter:       Class,  6-8-720. 

Volts,  125. 

Amperes,  64. 
Generator:  500  KW.  direct-connected. 

It  will  be  noticed  that  the  motors  of  the  new  machines  are  larger, 
and  are  specified  to  operate  at  80  per  cent  power-factor.  This  was 
necessary  to  allow  for  over-excitation  in  the  case  of  machines  of 
recent  design.  The  old  machines,  however,  will  stand  considerable 
over-excitation,  because  the  design  of  motors  was  formerly  more 
liberal  in  overload  capacity. 

A.  A.  M. 


INDEX 


Adams,  C.  A.,  resume"  of  paper  of,  225 
Advantages     and     disadvantages     of 

synchronous  motors,  140 
Air-gap,  length  of,  164 

local  corrections  of,  251 
Algebraical  relations  deduced  from  the 

diagram,  37 

Alternating  current,  starting  by,  106 
-field  motors,  xvi 

synchronous  motors,  145 
Alternators,     construction      of,     and 
application  of  calculation  of 
reactances,  267 
Mordey,  38,  47,  53 
ordinary,  polyphase  form  of,  9 
single-phase,  case  of,  264 
testing  according  to  the  theory  of 

two  reactions,  270 

Ampere-turns  in  the  case  of  unsatu- 
rated  armature,  diagram  of, 
241 

Armature,  case  of  a  saturated,  248  r 
reaction,  92 
self-inductance  of,  242 
reactions,  calculation  of,  236 
reduction    to  the    single-direct 

reaction,  174 

Annual  operating  cost,  saving  in,  75 
Arnold,  E.,  236 


B 


Bedell  and  Ryan,  103 

method    of    determining    phase- 
angle,  139 
Behn-Eschenburg  short-circuit  curves, 

132 

Bipolar  diagrams,  33 
principle  of,  32 


Blakesley's    method    of    representing 

operative  conditions,  20 
Blondel,  Prof.  Andre,  236 


Calculation  of  constants,  252 

of  reactions,  252 
Case  of  symmetrical  polyphase  motors, 

iQ 

Cassel,  power  transmission,  52 
Circuit,  or  line,  and  compensation,  65 
Classification     of     single-phase     and 

polyphase  motors,  xvi 
Comparison  between  synchronous  and 

induction-motors,  90 
of  outputs,  62 
Compensation,  economics  of,  72 

with  respect  to  the  generators,  86 

to  the  line  or  circuit,  65 
Complex   variables,    equation   of    the 
synchronous    motor    by    the 
method  of,  23 
Compound-excitation,  196 
Constant  field  motors,  xvi 

potential,  supply  system,  100 

V-curves  for,  200 
Constants,  calculations  of,  252 
Construction,  1-3 

Converter    operation,    representation 

of  with  constant  potential  at 

primary        terminals        and 

brushes,  180 

rotary,  characteristic  features  of, 

194 

lag-characteristics  of,  203 
special  applications  of,  223 
Cornu,  on  oscillations,  126 
Corrected  diagram,  first  application  of, 
95 


291 


292 


INDEX 


Cost,  annual  operating,  saving  in,  75 
of  equipment,  saving  in,  72 

Coupled    synchronous    motors,     dia- 
grams of,  166 

Crocker,  Dr.  F.  B.,  207 

Current-controllers,  109 
limit-circle  of,  37,  47 
-minimum,  existence  of,  55 
of  synchronous  motors,  effect  of, 

on  distribution-systems,  64 
-supply  voltage,  value  of,  183 
values,  182 

Curve  of  reaction  curve,  61 

Curves,  characteristic  test,  132 

indicating  power  developed,  44 

D 

Damper  of  Hutin  and  Leblanc,  126 
Damping  circuits,  221 
of  oscillations,  126 
Definition    of    synchronous    motors, 

xvii 

Determining  maximum  power,  51 
Determination  of  reactive  current,  95, 

198 
Determining  the  practical  stability  of 

synchronous  motors,  51 
Detroit  Edison  Co.,  on  power-factors, 

288 
Diagram,   corrected,   first  application 

of,  95 

fundamental,  177 
of   ampere-turns   in   the   case   of 

unsaturated  armature,  241 
of  E.  M.  F.'s  and  current  of  an 
alternator   with  .unsaturated 
armature  and  with  saturated 
field  magnet,  239 
of  the  first  kind,  applications  of, 

35 

of  the  first  kind,  use  of,  57 
of  the  second  kind,  40 
transformations,  149 
Diagrams,  simplified,  104 
Direct  current,  starting  by,  106 
Distributing  system,  economy  of  com- 
pensation for,  80 


Distribution-systems,  effect  on  of 
current  of  synchronous 
motors,  64 

-voltage,  regulation  of,  85 
Dynamometer,    Pronybrake    method, 

134 
torsion  type,  137 


Early  types  of  synchronous  motors, 

xviii 

Economy  of  compensation,  for  the  dis- 
tributing system,  80 
Efficiency,  measurement  of,  134 
Electric  current  supply  to  rotary  con- 
verters, 177 
Elementary    expansion   of    polyphase 

synchronous  motors,  9 
of    single-phase  synchronous 

motors,  14 
E.  M.  F.  diagrams,  150 

fluctuations  in  rotary  converters, 

220 

regulation  of  supply  by  com- 
pounding of  the  generator, 
210 

Empirical  coefficients,  255 
Equal  electromotive  forces,  case  of,  5 

phase,  lines  of,  36,  47 
Equation  of  the  synchronous  motor  by 
the  method  of  complex  vari- 
ables, 23 

Equations  of  synchronous  motors,  15 
Equipment,  saving  in  cost  of,  72 

table,  76 

Excitation,  different  values  of,  200 
compound,  196 
constant,    lines    of    equal    power 

occurring  with,  36 
of  synchronous  motors,  26 
separate,  117 

application  in  case  of,  206 
Experimental  properties,  5 

tests,  137 

Experiments  at  South  Foreland,  xvii 
Expression  for  reactive  current,  61 


INDEX 


293 


Ferraris,  Galileo,  145 

Field  due  to  a  commutated  current, 

121 
magnets,  stray  field  of,  242 

with  divided  winding,  252 
-saturation,  influence  of,  on  sta- 
bility, 101 
Fort      Wayne      Company,      starting 

method,  114 
method  of  current  commutating, 

30 

Forces,  equal  electromotive,  case  of,  5 
unequal  electromotive,  case  of,  7 
FresnePs  method  of  vectors,  20 
Fundamental  diagram,  177 


Generator,  compounding  of,  to  regu- 
late supply  of  E.  M.  F.,  210 
power-factor  of,  192 

Generators,  compensation  with  respect 
to,  86 

Gramme  commutators,  115 

Graphical  representation  of  operative 
conditions,  20 

Guilbert,  C.  F.,  236 

H 

Herdt,  L.  A.,  236 
Hobart  and  Punga,  236 
Hopkinson,  Dr.  J.,  15 

analytical  theory,  15 
Hutin  andLeblanc's  damper,  126 
Hutin,  on  oscillations,  131 


Induction    and    synchronous    motors, 
comparison  between,  90 

Influence   of   field-saturation  on   sta- 
bility, 1 01 
of  wave-form  of  E.  M.  F.,  102 

Initial  synchronizing,  theory  of,  117 


Janet,  M.,  24 


Kapp  type  of  synchronous  motors,  90 


Labour  starting  method,  113 

synchronous  motors,  51 
Lag  of  current,  variations  of  reactance 

with,  92 
Leblanc,  M.  Maurice,  14 

on  oscillations,  131 

starting  method,  113 
Length  of  air-gap,  164 
Limit-circle  of  current,  37 
Line  of    equal  power  occurring  with 

constant  excitation,  36 
Lines  of  equal  phase,  36,  47 

M 

Measurement  of  efficiency,  134 
Mordey  alternators,  38,  47,  53 


N 


Notation,  172 

for  study  of  operation,  31 
Numerical  examples,  53,  69,  75,  84,  88 


O 


Observations  on  the  E.  M.  F.,  induced 

at  the  poles,  108 
Oerlikon  type  of  motor,  133 
Operation  of  a  motor  with  constant 

excitation,  49 

with  constant  excitation,  100 
Operations,    factors    determining    the 

practical  condition  of,  176 
Operative    conditions,    representation 

of,  20 
Oscillations,  damping  of,  126 

of  converters  connected  in  parallel, 

220 

of  synchronous  motors,  122 
long-period,  130 
Outputs,  comparison  of,  62 
Overrating    of     motor-generator    sets 
to  improve  power-factor,  288 


294 


INDEX 


Parallel  working  of  rotary  converters, 

219 

Perot,  M.,  103 

Phase-angle,  determining,  139 
-converters,  224 
indicators,  108 
Picon,  M.  R.  V.,  236 
Polyphase  alternators,  ordinary  form 

of,  9 

motors,  symmetrical,  case  of,  19 
synchronous  motors,   elementary 

expansion  of,  9 

Power  developed,  curves  indicating,  44 
-factor  of  the  generator,  192 

improving,  287 
output,  50 

-values   as   function  of   the  lag- 
angle,  43 

Predetermination  of  V-curves,  57 
Principles  of   the   theory  of   two   re- 
actions, 237 
Prony-brake-dynamometer,  134 


R 


Reactance  coil,  use  of,  in 

value  of,  183 
Reactances,  calculation  of  applied  to 

alternator  construction,  267 
Reaction  synchronous  motors,  141 
Reactions,    practical    calculations    of, 

252 
two,  principles  of  the  theory  of, 

237 
Reactive      counter-ampere-turns      of 

armature  241 
current,  curve  of,  61 

determination  of,  95,  198 
expression  for,  61 
upper  limit  of,  202 
values    for    a    given    voltage 
variation  as  a  function  of  the 
load,  182 

Regulation  of  distribution-voltage,  85 
Revolving  field  motors,  xxi 


Rotary  converters  as  generators,  224 
as  partial  generators,  224 
conditions     of     electric     current 

supply  to,  177 

characteristic  features  of,  194 
eliminating    causes    of    irregular 

operation,  213 

for  transforming  direct  to  alter- 
nating current,  222 
lag  characteristics  of,  203 
parallel  working  of,  219 
speed  oscillations  in,  220 
under   load,    effective   character- 
istics of,  205 
Ryan  and  Bedell,  103  . 


Saving  in  annual  operating  cost,  75 

in  operating  cost,  table,  77 
Self-excitation,  30 

-inductance  of  armature,  242 
Separate  excitation,  117 
Series-excitation,  27 
Short-period  oscillations,  122 
Shunt-excitation,  28 

-winding,  possibility  of  suppress- 
ing, 211 

Single-phase  alternators,  case  of,  264 
machines,  starting,  112 
synchronous   motors,   elementary 

expansion  of,  14 

Soci6te"  PEclairage  Electrique,  type  of 
synchronous  motors,  table,  5 
Speed-oscillations      in     rotary      con- 
verters, 220 
Stability,  influence  of  field-saturation 

on,  101 
of  synchronous  operation,  n 

motors,  determining,  51 
variations  of,  with  operating  con- 
ditions, 52 

Starting  apparatus,  accessory,  108 
by  alternating  current,  106 
by  direct  current,  106 
of  single-phase  machines,  112 
Steinmetz  notation,  23 


INDEX 


295 


Swinburne's  "hedgehog"  transformers. 

xix 

Synchronism,  107 
Synchronous    and    induction    motors, 

comparison  between,  90 
operation,  stability  of,  1 1 
motors  with  alternating  fields,  145 
Symmetrical  polyphase  motors,   case 
of,  19 


Three-part  pole,  232 
Transverse  reaction,  method  of  deter- 
mining, 279 
Two-part  pole,  234 


U 


Unequal  electromotive  forces,  case  of,  7 


Tap  voltage,  231 

Table,  empirical  coefficients,  255,  258, 

259 
motors  of  the  Societe  PEclairage 

Electrique,  5 
numerical  example,  78 
saving  in  cost  of  equipment,  76 

in  operating  cost,  77 
single-phase  and  three-phase  syn- 
chronous motors  of  American 
manufacture,  285,  286 
theoretical  coefficients,  263 
Tesla  starting  method,  113 
Testing  alternators,  270 
Theoretical  coefficients,  263 
comparison  with,  261 
form  of  V-curves,  59 
Theory,    analytical,    of    synchronous 

motors,  15 
Thompson,  Prof.  S.  P.,  64 


Value  of  current-supply  voltage,  183 

of  reactance,  183 
Variation  of  voltage  by  the  split  pole, 

225 
Variations  of  stability  with  operating 

conditions,  52 
V-curves,  56 

for  constant  potential,  200 
predetermination  of,  57 
theoretical  form  of,  59 
Voltage    regulation    by    varying    the 
reactance  X  in   the  current, 
211 

of  at  terminals,  189,  190 
variation  by  the  split  pole,  225 


W 


"Watted"  current,  34 
Wave-form  of  E.  M.  F.,  influence  of, 
102 


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